Bicretieria Optimization in Routing Games

Bicriteria Optimization in Routing Games Costas Busc h Computer Science Department Louisiana State Univ ersit y 280 Coates Hall Baton Rouge, LA 70803 , USA busch@csc.lsu.e du Ra jgopal Kannan Computer Science Department Louisiana State Unive rsity 279 Coates Hall Baton Rouge, LA 7 0803, USA rk annan@csc.lsu.edu Abstract Two impo rtant metrics for measuring the quality of routing paths a re the maxim um edge c o n- gestion C and maximum path length D . Here, we study bicr iteria in r outing games where each play er i selﬁshly se le cts a path that simultaneously minimizes its maxim um e dg e c o ngestion C i and pa th length D i . W e study the stability and pr ice of anarch y of t wo bicriteria g a mes: • Max games , where the so cial cos t is max( C, D ) and the play er cost is max ( C i , D i ). W e prov e that ma x g ames are sta ble and c onv ergent under b es t- r esp onse dyna mics , and that the price of anarch y is bounded ab ov e b y the maximum path length in the play ers’ strateg y sets. W e also show tha t this bound is tight in worst-case scenario s. • Sum games , wher e the so cial cost is C + D and the play er cost is C i + D i . F or sum games, we ﬁrst s how the negative res ult that there are game instances that hav e no Nas h- equilibria. Therefore, w e examine a n approximate game ca lled the sum-bucket game that is always conv ergent (and therefore sta ble). W e show that the pr ice of anarch y in sum-buck et games is bo unded ab ov e by C ∗ · D ∗ / ( C ∗ + D ∗ ) (with a p oly-lo g factor ), whe r e C ∗ and D ∗ are the optimal co ordina ted co ng estion and path leng th. Th us, the sum-buck et game has typically sup e rior price of ana rch y b ounds than the max game. In fact, when either C ∗ or D ∗ is small (e.g. constant) the so c ia l cost of the Nash-equilibria is very clos e to the co or dinated optimal C ∗ + D ∗ (within a p oly-log factor). W e also show that the pr ice of anarch y b ound is tight for case s where b oth C ∗ and D ∗ are lar ge. 1 In tro du c tion Routing is a fundament al task in comm unication net works. Routing algorithms provi d e paths for pack ets that will b e sen t o ve r th e n et w ork. There are t wo metrics that quan tify the qualit y of the paths return ed b y a routing algorithm: the congestion C , wh ich is th e maxim um num b er of paths that use any edge in the n et w ork, and the maximum path length D . Assuming there is a p ac k et for eac h p ath, a lo wer b ound on the delive r y time of the pac ket s is Ω(max( C, D )) (alternativ ely , Ω( C + D )). Actually , th er e exist pac k et sc h eduling algorithms that give n the paths, they deliv er the p ac k ets along the paths in time close to optimal O (max( C, D )) (alternativ ely , O ( C + D )) [7, 17, 18, 23, 25]. Motiv ated by the selﬁsh b eha vior of en tities in communicatio n net works, we study routing games where eac h pac ket’ s path is con trolled indep enden tly by a selﬁsh pla y er. W e mo del games with N pla ye rs , where eac h pla yer h as to s elect a path fr om a source to a destination no de. The ob jectiv e of eac h p lay er i is to select a path that sim ultaneously minimizes tw o parameters: th e congestion C i , whic h is the maxim um n u mb er of paths that use any edge in pla y er i ’s path, and the path length D i . W e examine t wo kind s of games: max games , where the pla y er’s cost function is max( C i , D i ), and sum games , where the pla y er’s cost fu n ction is C i + D i . In eac h of these games, th e play er’s ob jectiv e is to selﬁsh ly minimize its cost in an unco ordinated manner. F rom the pla y er’s p oin t of view, th e m inimization of the sum or max cost functions are justiﬁed ob jectiv es, sin ce it is sho wn in [3] that p la y er i ’s pack et can b e deliv ered in time ˜ O ( C i + D i ) (alternativ ely , ˜ O (max( C i , D i )). A natural pr oblem is to determine th e eﬀect of the p lay ers ’ selﬁshness on the w elfare of the wh ole comm unication net wo r k . In the max and sum games, the w elfare of the n et w ork is measur ed with the so cial cost functions max( C, D ) and C + D , resp ect ively . The c hoice of these so cial cost functions is appr opriate since th ey determine the total time needed to d eliv er the pack ets repr esen ted b y the pla ye rs . W e examine the consequ ence of the selﬁsh b eha vior in Nash equilibria, which are stable states of the game w here no pla y er can unilaterally impro ve her situation. The eﬀect of selﬁshness is quantiﬁed w ith the pric e of anar chy ( P oA ) [16, 24], whic h expresses h o w m u ch larger is th e worst so cial cost in a Nash equilibrium compared to the so cial cost in the optimal co ordinated solution. W e study the existence of Nash equilibria and the p rice of anarc hy for max games and sum games, where we ﬁnd that these games pro du ce diﬀerent results with their o w n mer its. 1.1 Max Games First, w e examine max games (the so cial cost is max( C + D )). W e pro ve that ev ery max game has at least one Nash equilibrium. Th e equilibriu m can b e obtained b y b est resp onse d ynamics, where a pla y er greedily c hanges, whenever p ossible, the curr en t path to an alternativ e path with lo wer cost. With b est resp onse mo ves the game ev en tually conv erges to a Nash-equilibrium. W e sh ow that the optimal coord inated solution is a Nash-equilibr iu m to o. Th us, max games games hav e v ery go o d Nash equilibria. This observ ation is quanti ﬁ ed in terms of the pric e of stability ( P oS ) [1, 2] whic h expr esses how muc h larger is the b est so cial cost in a Nash equilibriu m with resp ect to the so cial cost in the optimal co ord in ated solution. Therefore in max games it holds that P oS = 1. W e then examine the worst Nash equilibr ia. W e b ound th e price of anarc hy ( P oA ) in max games w ith resp ect to the maxim um allo wable path length L for the pla yers in the n et w ork, and the num b er of no d es n in the graph : P oA = O ( L + log n ) . 1 W e prov e that this b ound is worst case optimal (within add itiv e terms). S p eciﬁcally , we provide an example game in a ring net wo rk where the optimal co ordin ated so cial cost is 1, while there is a Nash equilibrium w ith cost O ( L ) = O ( n ). 1.2 Sum Games W e contin ue with examining sum games (the so cial cost is C + D ). Intuiti vely , sum games hav e the p oten tial to giv e b ette r price of anarc hy th an the max games b ecause b oth parameters ( C and D ) aﬀect th e c hoices at all the time, eve n w hen one parameter is larger than the other. F or example, the ring game that w e mentioned ab o ve has price of anarch y equal to 1 in the sum game. Ho w eve r , w e prov e a limitation of sum games: not all sum games ha v e Nash equilibria; there exist instances of sum games w ith a small n u m b er of pla y ers that do not hav e Nash equilibria at all. This limitation directed us to wards exploring alternativ e games w h ic h are stable (hav e Nash equilibria) and hav e similar c h aracteristics with the original sum games. W e foun d such a game v ariation that w e call sum-buck e t game . In su m-buck et games the pla y ers are d ivided into log n classes, called buck ets , according to the pac ke t paths that they choose. Buc ke t k holds the paths of p la y ers with length in range [2 k , 2 k − 1 ). S upp ose that pla ye r i ’s path is in b uc ke t k . The normalize d c ongestion of pla yer i , denoted C i , is measured with r esp ect to th e p aths that b elong to buc ke t k . The normalize d length of play er i ’s path is D i = 2 k − 1 − 1, (wh ich is a facto r 2 appro ximation of the original length). Pla yer i ’s cost function is C i + D i . Thus, in su m-buc ket games only play ers in the same buc ket comp ete with eac h other, while pla y ers in diﬀerent buc ket s do not interfere. The normalize d so cial c ost function is deﬁn ed to b e C + D , wh ere C is the maxim um normalized congestion in an y buc ket, and D is the m axim um n ormalized depth of all paths. W e ﬁrst sho w that sum-buck et games alw a ys hav e Nash equilibr ia, whic h can b e obtained with b est r esp onse dynamics. W e then examine the qualit y of the Nash equilibria. F or ev ery game there is a corresp onding co ordinated buck et r outing problem. W e can b ound the price of anarch y P oA w ith resp ect to the normalized optimal congestion C ∗ and normalized p ath length D ∗ in the optimal co ordinated solution. W e obtain: P oA = O C ∗ · D ∗ C ∗ + D ∗ · log 2 n ! Therefore, w h en either of C ∗ or D ∗ is small (e.g. a constant), the Nash equilibriu m p ro vides a v ery go o d appr o ximation (within a p oly-log factor from optimal) to the u nco ordinated routing problem. In suc h s cenarios, the pr ice of selﬁsh ness is sm all. How ev er, when b oth C ∗ or D ∗ are sim u ltaneously large, the appr o ximation b ecomes worse (though still typical ly lo wer th an th e P oA of the max game since the P oA is b ounded by the smaller of C ∗ or D ∗ , wh er e D ∗ < 2 L ). Nev erth eless, even in these scenarios the PoA b ound is tight in certain games instances. Sum-bu c k et games are in teresting v ariations of sum games b ecause they are stable, and they can b e used to appr o ximate solutions f or the C + D so cial cost. F or any s u m-buck et game, there is a corresp onding “original” co ordinated rou tin g problem where the ob jectiv e is to m inimize the so cial cost C + D without usin g b uc ke ts. It holds th at C ≤ C ≤ C · log n and D ≤ D ≤ 2 D ; th us, C + D = O ( C · log n + D ). In other w ords, the normalized so cial cost can b e used as an appro ximation for the “origi n al” so cial cost. Let P oA ′ denote ho w m uch larger is the w orst equilibrium of a s u m-buck et game compared to the optimal solution of the co ordin ated original problem (with resp ect to the so cial cost C + D ). It holds that P oA ′ ≤ P oA · log n . Consequently , 2 the observ ations that w e made ab o ve for the P oA in sum-b uc ke t games apply also with resp ect to the original r outing problem. F or example, wh en one of C ∗ or D ∗ is small (e.g. a constan t), then the Nash equ ilibrium of the sum -buc ket game pro vides a v ery goo d approxi mation (within a p oly log factor fr om optimal) to th e co ord inated original r outing problem. 1.3 Related W ork Routing games (on congestion) w ere in tro d uced and studied in [22, 26]. The notion of price of anarc hy wa s introdu ced in [16]. Since then , many routing game mo d els hav e b een stud ied whic h are distinguish ed by the top ology of the net work, cost functions, t yp e of traﬃc (atomic or splittable), nature of strategy sets, and kind of equilibria (pu re or mixed). S p eciﬁcally , pu r e equilibria with atomic ﬂ ow ha ve b een studied in [4, 5, 19, 26, 31] (our w ork ﬁts into this category), and with splittable ﬂo w in [27, 28, 29, 30]. Mixed equilibria with atomic ﬂow h a v e b een stud ied in [8, 9, 10, 12, 13, 14, 15, 16, 20, 21, 24], and with sp littable ﬂo w in [6, 11]. T o our kno wledge there is no previous work that considers routing games that optimize tw o criteria sim u ltaneously . Most of th e w ork in the literature uses a single cost metric whic h is related to the congestion. A common metric f or the p la y er cost is the sum of the congestions on all the edges of the pla y er’s path (w e denote this kind of play er cost as p c ′ ) and th e resp ectiv e so cial cost is the cost of the worst p la y er’s path (w e denote this so cial cost as S C ′ ) [5, 14, 28, 29, 30, 31]. Ho w ever, as we discus sed b efore, in pack et scheduling algorithms, the pc ′ or S C ′ do not go vern the pac k et dela ys; max( C i , D i ) or C i + D i go v ern the pac k et dela y . Other combinations of pla yer costs and so cia l cost s h a v e b een studied in the literature: p la y er cost pc ′ and so cial cost C has b een studied in [5, 6, 8, 9, 10, 11, 12, 15, 16, 21, 24, 27]; pla yer cost C i and so cial cost C has b een stu died in [4]; other v ariations hav e b een stud ied in [13, 19, 20, 26]. The v ast ma jorit y of the w ork on routing games has b een p erformed for parallel link net works, with only a f ew exceptions on general net work top ologies [4, 5, 6, 27]. Our work is closer to [4]. W e extended some results present ed in [4 ] to apply to bicriteria, instead of the single criterium of congestion, pla y er cost C i and so cial cost C , that w as u sed in [4 ]. Sp eciﬁca lly , the particular tec hn iques that w e use to pro ve existence of Nash equilibr ia w ith b est resp onse d ynamics, and also to pr o v e u p p er b ounds on the price of anarch y , were originally in tro d uced in [4]. Here, w e mo diﬁed and extended appropriately these tec hn iques in a non-trivial w a y to apply to our new cost functions. Outline of Pa p er W e pro ceed as follo ws. In Section 2 w e giv e basic deﬁnitions. W e study max games in Section 3 and s um games in Section 4. W e ﬁn ish with our conclusions in Section 5. Due to sp ace limitations some pro ofs hav e mov ed to the ap p endix. 2 Deﬁnitions An ins tance of a r outing game is a tuple R = ( N , G, P ), where N = { 1 , 2 , . . . , N } are the pla yers, G = ( V , E ) is a graph with n o des V and edges E , and the graph has paths P = S i ∈ N P i , where P i is a collectio n of a v ailable paths in G for play er i . Eac h p ath in P i is a path in G that has the same source u i ∈ V and destinatio n v i ∈ V ; eac h path in P i is a pur e str ate gy a v ailable to play er i . A pur e str ate gy pr oﬁle p = [ p 1 , p 2 , · · · , p N ] is a collection of pure strategies (paths), one for eac h 3 pla ye r, where p i ∈ P i . W e refer to a pure strategy pr oﬁle as a r outing . On a ﬁnite n et w ork, a routing game is n ecessarily a ﬁn ite game. F or an y routing p and an y edge e ∈ E , th e e dge-c ongestion C e ( p ) is the num b er of paths in p that use edge e . F or an y path p , the p ath-c ongestion C p ( p ) is the maxim um edge congestion o v er all edges in p , C p ( p ) = max e ∈ p C e ( p ). W e will u s e the notation C i ( p ) = C p i ( p ), f or any user i . T he network c ongestion is the maxim u m ed ge-conge stion o ve r all edges in E , that is, C ( p ) = max e ∈ E C e ( p ). W e denote the length (num b er of edges) of any path p as | p | . F or an y user i , we will also u se the notation D p i ( p ) or D i ( p ) to d enote the length | p i | . The longest p ath leng th in P is denoted L ( P ) = max p ∈P | p | . W e will d enote by D ( p ) the maxim um path length in routing p , that is D ( p ) = m ax p ∈ p | p | . When the con text is clear, we will d rop th e dep enden ce on p and R and use the n otation C e , C p , C i , C, L, D p , D i , D . F or game R and routing p , the so cial c ost (or glob al c ost ) is a function of routing p , and it is denoted S C ( p ). The player or lo c al c ost is also a function on p d enoted p c i ( p ). W e use the standard notation p − i to refer to the collect ion of paths { p 1 , · · · , p i − 1 , p i +1 , · · · , p N } , and ( p i ; p − i ) as an alternativ e notation for p whic h emphasizes the dep endence on p i . Pla yer i is lo c al ly optimal in routing p if pc i ( p ) ≤ pc i ( p ′ i ; p − i ) for all paths p ′ i ∈ P i . A routing p is in a Nash Equilibriu m (w e sa y p is a Nash-r outing ) if ev ery play er is lo cally optimal. Nash-routings quant ify the notion of a stable selﬁsh outcome. A routing p ∗ is an optimal pur e strategy p roﬁle if it has minim um attainable so cial cost: for an y other p ure strategy pr oﬁle p , S C ( p ∗ ) ≤ S C ( p ). W e quantify the qualit y of the Nash-routings by the pric e of anar chy ( P oA ) (sometimes r eferred to as the co ordination ratio) and the pric e of stability ( P oS ). Let P d enote the s et of distinct Nash- routings, and let S C ∗ denote the so cial cost of an optimal routing p ∗ . T hen, P oS = inf p ∈ P S C ( p ) S C ∗ , P oA = sup p ∈ P S C ( p ) S C ∗ . 3 Max Games Let R = ( N , G, P ) a routing game suc h that for an y routing p the so cial cost f unction is S C ( p ) = max( C ( p ) , D ( p )), and the play er cost f unction p c i ( p ) = max( C i ( p ) , D i ( p )). W e refer to suc h routing games as max games . First, we sho w that max games hav e Nash-routings and the price of stabilit y is 1. Then, w e b ound the p rice of anarc hy . 3.1 Existence of Nash-routings in Max Games W e sho w th at max games ha v e Nash-routings. W e pro ve this result by ﬁrst giving a totaly order for the rou tin gs using a form of lexico graph ic order in g. Then w e sho w th at an y greedy mo ve of a p la y er can only giv e a new routin g with smaller order. Thus, the greedy mo ve s w ill con v erge either to the smallest r outing or to a routing where no pla ye r can improv e furth er. In either case, a Nash-routing will b e reac hed. Let R = ( N , G, P ) b e a max r ou tin g game. Let r = max( N , L ). F or any routing p w e deﬁn e the r outing ve ctor M ( p ) = [ m 1 ( p ) , . . . , m r ( p )], wh ere m i ( p ) = a i ( p ) + b i ( p ), and a i ( p ) is the num b er of paths with congestion i , and b i ( p ) is the n umb er of paths with length i . Note that if S C ( p ) = k then m k 6 = 0 and m ′ k = 0 f or all k ′ > k . W e d eﬁne a total order on the routings as follo w s. Let p and p ′ b e t wo routings, with M ( p ) = [ m 1 , . . . , m r ], and M ( p ′ ) = [ m ′ 1 , . . . , m ′ r ]. W e sa y that M ( p ) = M ( p ′ ) if m i = m ′ i for all 1 ≤ i ≤ r . 4 W e say that M ( p ) < M ( p ′ ) if th ere is a j , 1 ≤ j ≤ r , such that m k = m ′ k for all k > j , and m j < m ′ j . W e ord er the p and p ′ according to th e order of their resp ectiv e v ectors, that is p ≤ p ′ if and only if M ( p ) ≤ M ( p ′ ). Note that for any t wo p and p ′ it either holds that p = p ′ or p < p ′ . That is, the routings are totally ordered. Consider an arbitrary routing p . If p is not a Nash-routin g, there is at least one u ser i w hic h is not lo cally optimal. Then a gr e e dy move is av ailable to play er i in wh ic h the play er can obtain lo w er cost b y changing the path from p i to some other path p ′ i with lo wer cost. In other words, the greedy mo v e tak es the original routing p = ( p i ; p − i ) to a routing p ′ = ( p ′ i ; p ′ − i ) with impro ved pla ye r cost pc i ( p ′ ) < pc i ( p ), su c h that p i is replaced by p ′ i and th e remaining paths sta y the same ( p − i = p ′ − i ). W e sho w no w that an y greedy mo ve giv es a smaller ord er routing: Lemma 3.1 If a gr e e dy move b y any player takes a r outing p to a new r outing p ′ , then p ′ < p . Pro of: Let pc i ( p ) = max( C i ( p ) , D i ( p )) = k max 1 , and min( C i ( p ) , D i ( p )) = k min 1 (clearly , k max 1 ≥ k min 1 ). Let also pc i ( p ′ ) = max( C i ( p ′ ) , D i ( p ′ )) = k max 2 , and min( C i ( p ′ ) , D i ( p ′ )) = k min 2 (clearly , k max 2 ≥ k min 2 ). Since pla y er i can decrease its cost in p ′ , k max 2 < k max 1 . Consider no w the vec tors M ( p ) = [ m 1 , . . . , m r ] and M ( p ′ ) = [ m ′ 1 , . . . , m ′ r ]. These tw o ve ctors are the same except p ossibly for en tries k max 1 , k min 1 , k max 2 , k min 2 , whic h corresp ond to the p ositions that are aﬀected by p aths p i and p ′ i . It holds that m k max 1 > m ′ k max 1 , since when the path switches to p ′ i , m k max 1 = a k max 1 + b k max 1 decreases by at least one b ecause either a k max 1 decreases by one (if the new path h as lo wer congestion) or b k max 1 decreases by on e (if the new p ath has low er length). Since k max 2 < k max 1 , M ( p ) > M ( p ′ ) implying that p > p ′ . Since th er e are only a ﬁ nite num b er of routings, Lemma 3.1 implies that starting from arb itrary initial state, ev ery b est resp onse d y n amic con v erges in a ﬁn ite time to a Nash-routing, where ev ery pla ye r is lo cally optimal. Since the routings are tota lly ordered, there is a routing p min whic h is the minimum, that is, f or all routin gs p , p min ≤ p . Clearly , the minimum routing is also a Nash- routing. The minim um routing p min ac hiev es also optimal social cost, since if there w as another routing p ′ with lo w er so cial cost, then it can b e easily sh o wn th at p ′ < p min , wh ic h is con tradiction. Th u s, the price of s tabilit y is 1. Therefore, we h a v e the follo wing result: Theorem 3.2 (St abilit y of max games) F or any max g ame R , every b est r esp onse dynamic c onver ges to a Nash-r outing, and the pric e of stability is P oS = 1 . 3.2 Price of Anarch y in Max Games W e b ound the price of anarc hy in max games. Cons id er a max routing game R = ( N , G, P ), where G h as n no des. Theorem 3.2 imp lies th at there is at least one Nash-routing. Consider a Nash- routing p . Denote C = C ( p ) and D = D ( p ). Let p ∗ b e the optimum (coordin ated) routing with minim u m so cial cost. Denote C ∗ = C ( p ∗ ) and D ∗ = D ( p ∗ ). Note that eac h pa yer i ∈ N h as a p ath p i ∈ p and a corresp ond ing “optimal” path p ∗ i ∈ p ∗ from the play er’s source to th e destination. F or eac h edge e ∈ G , d enote Π e ( p ) the set of pla y ers wh ose paths in routing p use edge e . W e deﬁne H to b e a set that con tains all edges e ∈ G with congestion C e ( p ) ≥ D + 2. Consider an edge e ∈ H . L et i ∈ Π e ( p ) b e a pla y er whose p ath p i in routing p uses edge e . W e d eﬁne f ( e, i ) to b e a set th at con tains all edges e ′ ∈ p ∗ i with C e ′ ( p ) ≥ C e ( p ) − 1. It holds that | f ( e, i ) | ≥ 1, since in routing p play er i p r efers path p i instead of p ∗ i b ecause th ere is at least one edge e ′ ∈ p ∗ i 5 with C e ′ ( p ) ≥ C e ( p ) − 1 > D . Let f ( e ) = ∪ i ∈ Π e ( p ) f ( e, i ). F or any set of edges X ⊂ H , w e deﬁ ne f ( X ) = S e ∈ X f ( e ). Lemma 3.3 L et Z b e the set that c ontains al l e dges e with c ongestion C e ( p ) ≥ C − 2 log n . If C ≥ D + 2 lg n + 2 , then ther e is a set of e dges X ⊆ Z suc h that | f ( X ) | ≤ 2 | X | . Pro of: W e r ecursiv ely construct a sets of edges E 0 , . . . , E 2 lg n , s uc h that E i = E i − 1 ∪ f ( E i − 1 ), and set E 0 con tains all the edges e with congesti on C e ( p ) = C . F rom the construction of those sets it holds that for an y e ∈ E j , C e ( p ) ≥ C − j , where 0 ≤ j ≤ 2 lg n . Th us, E j ⊆ Z , for all 0 ≤ j ≤ 2 lg n . (Note that Z ⊆ H .) W e can show th at there is a j , 0 ≤ j ≤ 2 lg n , su c h th at | f ( E j ) | ≤ 2 | E j | . Supp ose for contra- diction th at su c h a j do es not exist. Thus f or all j , 0 ≤ j ≤ 2 lg n , it holds that | f ( E j ) | > 2 | E j | . In this case, it it straigh tforwa r d to sho w that | E k | > 2 | E k − 1 | , for an y 1 ≤ k ≤ 2 lg n . S ince | E 0 | ≥ 1, it holds that | E 2 lg n | > 2 2 lg n = n 2 . Ho we ver, this is a con tradiction, since the num b er of edges in G do not exceed n 2 . Lemma 3.4 If C ≥ D + 2 lg n + 2 , then C < 2 LC ∗ + 2 lg n . Pro of: F rom Lemma 3.3, there is a set of edges X ⊂ Z w ith | f ( X ) | ≤ 2 | X | . F or eac h e ∈ Z it holds th at C e ( p ) ≥ C − 2 lg n . Let M = P e ∈ X C e ( p ) ≥ | X | ( C − 2 lg n ), where M denotes the total utilization of the edges in X b y th e p aths of the pla yers in Π. Let Π = S e ∈ X Π e ( p ), that is, Π is the set of pla ye r s whic h in r outing p their paths use edges in X . By constru ction, the congestion in routing p in eac h of the edges of X is caused only b y the p la y ers in Π. Sin ce path lengths are at most L , eac h pla y er in Π can use at most L edges in X . Hence, M ≤ L · | Π | . Cons equen tly , | X | ( C − 2 lg n ) ≤ L · | Π | , wh ic h giv es: C ≤ ( L · | Π | ) / | X | + 2 lg n . By the d eﬁnition of f ( X ), in the optimal routing p ∗ eac h user in Π has to use at least one edge in f ( X ). Thus, edges in th e optimal routing p ∗ , the edges in f ( X ) are used at least Π times. Th u s, there is some ed ge e ∈ f ( X ) with C e ( p ∗ ) ≥ | Π | / | f ( X ) | . Therefore, C ∗ ≥ | Π | / | f ( X ) | . Since | f ( X ) | ≤ 2 | X | , we obtain | Π | ≤ 2 C ∗ · | X | . Therefore: C ≤ 2 LC ∗ + 2 lg n . Theorem 3.5 (Price of anarc hy in max games) F or any max game R it holds that P oA = O ( L + log n ) . Pro of: Supp ose that p is the worst Nash-r outing with maximum so cial cost. W e ha ve P oA = S C ( p ) /S C ( p ∗ ). If C ≥ D + 2 lg n + 2, then S C ( p ) = C . F rom Theorem 3.4, P oA ≤ C / max( C ∗ , D ∗ ) ≤ (2 LC ∗ + 2 lg n ) /C ∗ ≤ 2 L + 2 lg n . If C < D + 2 lg n + 2, then S C ( p ) < D + 2 lg n + 2; thus P oA ≤ L + 2 lg n + 2. Hence, in b oth cases P oA = O ( L + log n ). There is a max game th at shows that the result of Theorem 3.5 is tigh t in the wo r st case. Consider a ring net wo r k with n no des and n edges. Giv e the same orienta tion to the edges, so that eac h edge has one left n o de and one right no d e. F or eac h edge e i , there is a corresp onding play er i whose source is the left n o de and the d estination is th e right no de of the edge. T he strategy s et of eac h pla ye r h as t w o paths: p ath p i whic h is only th e edge e i , and path p ′ i whic h go es around the ring. Note that p = [ p 1 , p 2 , . . . , p n ] is a Nash-routing with so cial cost 1. Ho wev er, p ′ = [ p 1 , p 2 , . . . , p n ] is also a Nash rou tin g with so cial cost n − 1. Thus, the price of anarch y is O ( n ) = O ( L ). 6 4 Sum Games Let R = ( N , G, P ) b e a routing game suc h that for any routing p the so cial cost function is S C ( p ) = C ( p ) + D ( p ), and the pla y er cost function pc i ( p ) = C i ( p ) + D i ( p ). W e refer to suc h routing games as sum games . W e ﬁrst s h o w th at such games ha ve instances without Nash-routings. Then w e describ e a v ariation of sum games, that w e call sum-bu ck et games, w hic h are stable and their equilibria h a v e go o d p rop erties. 4.1 A Sum Game w it hout Nash-routings Theorem 4.1 Ther e is sum game instanc e R = ( N , G, P ) that has no Nash-r outing. Pro of: The graph G is d ep icted in the ﬁgure b elo w. There are sev en pla yers, namely , N = { 1 , . . . , 7 } . Pla ye r s 1, 2, and 3, hav e r esp ectiv e str ategy sets P 1 = { p 1 , p ′ 1 } , P 2 = { p 2 , p ′ 2 } , and P 3 = { p 3 , p ′ 3 } . I n the ﬁgure for pla ye r i = 1 , . . . , 3 the resp ectiv e source and d estination no des are u i and v i . There are six critical edges denoted e 1 , . . . , e 6 that the paths use and which are sho w n in the ﬁgure as straight horizon tal lines. Th ese edges ma y ha ve congestion larger than 1. The squiggly part of the p aths are assumed to ha v e congest ion 1 and their length and their lengths are c hosen so that th e f ollo w ing relations hold: | p ′ 1 | = | p 1 | − 2, | p ′ 2 | = | p 2 | + 3, an d | p ′ 3 | = | p 3 | + 3 . p 1 p 1 p ′ 1 p ′ 1 p 2 p 2 p 3 p 3 p ′ 3 p ′ 3 p ′ 2 p ′ 2 p 1 p ′ 3 p 1 p 2 p 3 p 1 p ′ 2 p ′ 1 p 2 p ′ 1 p 3 u 1 u 2 u 3 e 1 e 2 e 3 e 4 (+1) v 1 e 6 e 5 p ′ 1 (+3) (+3) (+4) v 2 v 3 p 2 p 3 Pla y ers 4 to 7 are “p assive” in the sense that they ha v e only one p ath in their strateg y sets, and their sole purp ose is to create additional congestion on edges e 3 , e 4 , e 5 , e 6 (the paths of these pla ye rs are n ot shown explicitly in the ﬁgure). In particular, the passiv e p la y ers cause additional congestion 1 to edge e 3 , additional congestion 3 to e 4 and e 5 , and additional congestion 4 to e 6 . The additional congestion is depicted in the ﬁgure in side a parent h esis und er eac h edge. Since the only “activ e” pla ye rs are 1, 2, and 3, and eac h pla yer has t wo path c hoices, there are eigh t p ossible d iﬀeren t routings. W e examine eac h r outing and pro v e that it is not a Nash-r outing. W e use th e v ector [ p 1 , p 2 , p 3 ] to denote a routing wh ere the i th p ositio n of th e vecto r conta ins the path choice of user i . By setting explicit v alues to the p ath lengths, and computing the p la y er costs in eac h rou tin g, w e ﬁnd that: pla y er 1 is not lo cally optimal in r outings [ p 1 , p 2 , p 3 ] and [ p ′ 1 , p ′ 2 , p ′ 3 ]; pla ye r 2 is not lo cally optimal in routings [ p ′ 1 , p 2 , p 3 ], [ p 1 , p ′ 2 , p ′ 3 ], [ p 1 , p ′ 2 , p 3 ], and [ p ′ 1 , p 2 , p ′ 3 ]; and pla ye r 3 is not lo cally optimal in routings [ p ′ 1 , p ′ 2 , p 3 ] and [ p 1 , p 2 , p ′ 3 ]. 4.2 Sum-buc ket Games Here we describ e sum-bucket games , whic h are v ariation of sum games that are stable and their equilibria ha ve go o d prop erties. Let R = ( N , G, P ) d enote a sum-buc ket r ou tin g game. Th e paths 7 in P are d ivided into buckets B 0 , . . . , B lg L so that B k is a set that con tains all paths wh ose lengths are in r ange [2 k , 2 k +1 ). (W e use lg L ins tead of ⌈ lg L ⌉ to a vo id notatio nal clutter.) F or any p ath p ∈ P , let B ( p ) denote the in d ex of the b uc ket that p b elongs to; namely , if p ∈ B k , then B ( p ) = k . Consider no w a routing p . W e deﬁn e the normalize d length of path p ∈ p as D p ( p ) = 2 B ( p )+1 − 1 (whic h is the maximum p ossib le path length in buck et B ( p )). Note that all the paths in the same buc ket ha ve the same normalized length. F or any path p we deﬁ n e th e normalize d c ongestion C p ( p ) to b e the maxim um congestion on an y of th e edges of path p whic h is caused by the paths of r outing p whic h b elong to b uc ket B ( p ). Pla y er i ’s cost is pc i ( p ) = C i ( p ) + D i ( p ). Th e pla ye r is allo we d to switc h from one bu c k et to another. The normalized congesti on of p is C ( p ) = max p i ∈ p C i ( p ), and the norm alized length of p is D ( p ) = max p i ∈ p D i ( p ). Th e so cial cost of game R is S C ( p ) = C ( p ) + D ( p ). Belo w, w e s h o w that sum-b u c ke t games are stable an d then we b ound the price of anarch y . 4.2.1 Existence of Nash-routings in Sum-buck et Games Here we sho w that sum-bu c ke t games hav e Nash-routings. W e use the same tec hnique as in S ec- tion 3.1, where w e order the routings and pro ve that greedy mov es giv e smaller order r outings. Let R = ( N , G, P ) b e a sum-buck et routing game. L et r = N + 2 L − 1 (this is the maxim um p ossible pla ye r cost). F or any r outing p w e deﬁne the r outing ve ctor M ( p ) = [ m 1 ( p ) , . . . , m r ( p )] such that m j ( p ) is th e n u mb er of paths in p with cost j . W e deﬁn e a total ord er on the routings, with r esp ect to their vecto rs, in exactly the same wa y as wee did for the max games in Section 3.1. Using similar tec hniques as in the max games, w e can prov e that if a greedy mov e b y pla y er i tak es a routing p to a new routing p ′ , then p ′ < p . Since there are only a ﬁnite num b er of routings, eve r y b est resp onse dynamic con v erges in a ﬁnite time to a Nash-routing. Th erefore, w e get: Theorem 4.2 (St abilit y of sum-buc k et games) F or any sum-buc k et g ame R , every b est r e- sp onse dynamic c onver ge s to a N ash-r outing. 4.2.2 Price of Anarch y in Sum-buck et Games F rom Theorem 4.2, eve r y sum-buck et game has at least one Nash-routing. Here, we b ound the pr ice of anarc h y . Cons ider a sum-bu c k et routing game R = ( N , G, P ), where G h as n no d es. Consider a Nash-routing p . Denote C = C ( p ) and D = D ( p ). Let p ∗ b e the optimum (co ordin ated) routing with min im um so cial cost. Denote C ∗ = C ( p ∗ ) and D ∗ = D ( p ∗ ). Note that eac h pay er i ∈ N has a path in p i ∈ p an d a corresp ondin g “optimal” path p ∗ i ∈ p ∗ from the pla yer’s source to th e destination no d e. F or an y pla yer i , let s i denote the shortest path in P wh ic h connects the sour ce and destinations n o des of i . W e no w relate the paths lengths with the congestion: Lemma 4.3 In Nash-r outing p , for any player i with C i ≥ C − x , wher e x ≥ 0 , it holds that | p i | ≤ | s i | + x + 1 . Pro of: Supp ose for the sake of con tradiction that there is a play er i with | p i | > | s i | + x + 1. Then, pc i = C i + D i ≥ C − x + D i ≥ C − x + | p i | > C − x + | s i | + x + 1 = C + | s i | + 1 . Clearly , | s i | ≤ 2 B ( s i )+1 − 1. I f user i was to switch to p ath s i its cost wo u ld b e p c ′ i ≤ C + 1 + 2 B ( s i )+1 − 1, since the normalized length of s i is 2 B ( s i )+1 − 1, and s i has normalized congestion at most C b efore pla ye r i switc hes, and the normalized congestion of path s i increases to at most C + 1 after p la y er i switc hes to it. Therefore, pc ′ i ≤ C + 1 + | s i | < pc i . Thus, in p p la y er i w ould not b e optimal, which is a con tradiction, sin ce p is a Nash-routing. Therefore, | p i | ≤ | s i | + x + 1, as needed. 8 F or eac h edge e ∈ G let Π e ( p ) denote the set of pla y ers wh ose paths in rou tin g p u s e edge e . Let C e,p i ( p ) denote the num b er of pac k ets th at use edge e in p and are in the same b uc ke t as p i . Let C e ( p ) = max p i C e,p i ( p ) d enote the normalized congestion of edge e . F or any edge e , w e deﬁne f ( e, i ) to b e a set that con tains all edges e ′ ∈ p ∗ i with C e ′ ,p ∗ i ( p ) ≥ C e,p i ( p ) − D ∗ . L et f ( e ) = ∪ i ∈ Π e ( p ) f ( e, i ), and f or an y set of edges X , f ( X ) = S e ∈ X f ( e ). It can b e shown that in Nash-routing p it holds | f ( e, i ) | ≥ 1. Lemma 4.4 L et Z b e the set that c ontains al l e dges e with c ongestion C e ( p ) ≥ C − 2 D ∗ · lg n . Ther e is a set of e dges X ⊆ Z with | f ( X ) | ≤ 2 | X | . Pro of: W e r ecursiv ely construct a sets of edges E 0 , . . . , E 2 lg n , s uc h that E i = E i − 1 ∪ f ( E i − 1 ), and set E 0 con tains all the edges e with congestion C e ( p ) = C . W e can sho w that th ere is a j , 0 ≤ j ≤ 2 lg n , su ch that | f ( E j ) | ≤ 2 | E j | . Sup p ose for cont rad iction that suc h a j do es not exist. Th u s for all j , 0 ≤ j ≤ 2 lg n , it holds that | f ( E j ) | > 2 | E j | . In this case, it it straigh tforw ard to sho w that | E k | > 2 | E k − 1 | , for an y 1 ≤ k ≤ 2 lg n . S in ce | E 0 | ≥ 1, it holds that | E 2 lg n | > 2 2 lg n = n 2 . Ho w ever, this is a con tradiction, sin ce the num b er of edges in G cannot exceed n 2 . Th us, there is a j with | f ( E j ) | > 2 | E j | . W e will set X = E j . It only remains to sho w that X ⊆ Z . It suﬃces to sh o w that for an y E k and e ∈ E k , C e ( p ) ≥ C − k D ∗ , where 0 ≤ k ≤ 2 lg n . W e pro ve this b y induction on k . F or k = 0 w e hav e that ev ery edge in e has C e ( p ) = C = C − 0 · D ∗ , th us the claim trivially holds. F or the induction hyp othesis, supp ose th at the claim holds for an y k = t < 2 log n . In the induction step w e will prov e that the claim holds also for k = t + 1. W e hav e that E t +1 = E t ∪ f ( E t ). By induction hyp othesis, for an y e ∈ E t , C e ( p ) ≥ C − tD ∗ (th us, f or any e ∈ E t it holds that C e ( p ) ≥ C − ( t + 1) D ∗ )). Thus, for any e ∈ E k there is at least one path p i with C e,p i ( p ) ≥ C − tD ∗ . By deﬁnition of f ( e, i ), ev ery edge e ∈ f ( e, i ) has the p rop erty that C e ′ ,p ∗ i ( p ) ≥ C e,p i ( p ) − D ∗ ≥ C − ( t + 1) D ∗ . Thus, th ere is at least one path p ′ ∈ p whic h is in the same buck et with p ∗ i (namely , B ( p ′ ) = B ( p ∗ i )) for whic h it holds that C e ′ ,p ′ ( p ) = C e ′ ,p ∗ i ( p ). Therefore, C e ′ ( p ) ≥ C − ( t + 1) D ∗ . Consequently , f rom th e deﬁn ition of f ( E t ) it follo ws that for any edge e ′ ∈ f ( E t ) it holds that C e ′ ( p ) ≥ C − ( t + 1) D ∗ . By consid er in g the un ion of E t ∪ f ( E t ), we ha v e that the claim h olds for any edge in e ∈ E t +1 , as needed. Lemma 4.5 In N ash-r outing p it holds tha t C ≤ 18 C ∗ · D ∗ · lg 2 n . Pro of: F rom Lemma 4.4, there is a set of edges X ⊂ Z w ith | f ( X ) | ≤ 2 | X | . F or eac h e ∈ Z it holds that C e ( p ) ≥ C − 2 D ∗ · lg n . Let Π = S e ∈ X Π e ( p ), that is, Π is the set of pla yers whic h in routing p their paths us e edges in X . Let M = P e ∈ X C e ( p ), which denotes the total “normalized” utilization of th e edges in X . W e h a v e that M ≥ | X | ( C − 2 D ∗ · lg n ). By construction, the congestion in routing p in eac h of the edges of X is caused only b y the pla ye rs in Π. W e can b oun d the path lengths of the pla y ers in Π with resp ect to D ∗ as follo ws. Consider a pla ye r i ∈ Π. W e ha ve th at C i ( p ) ≥ C − 2 ¯ D ∗ · lg n . F rom Lemma 4.3 and the fact that | s i | ≤ D ∗ , we obtain: | p i | ≤ | s i | + 2 D ∗ · lg n + 1 ≤ D ∗ + 2 D ∗ · lg n + 1 ≤ 4 D ∗ · lg n. T h u s , the path length of eve ry pla yer in Π is at most K = 4 D ∗ · lg n . M can also b e b ounded as M ≤ K | Π | . Consequently , | X | ( C − 2 D ∗ · lg n ) ≤ K | Π | , whic h giv es: C ≤ K | Π | | X | + 2 D ∗ · lg n = 4 D ∗ · lg n · | Π | | X | + 2 D ∗ · lg n. (1) Since | f ( e, i ) | ≥ 1, in the optimal routing p ∗ the path of eac h user in Π has to use at least one edge in f ( X ). Thus, in the optimal routing p ∗ , the edges in f ( X ) are used at least | Π | times. Thus, there 9 is some edge e ∈ f ( X ) which in p ∗ is u sed b y at least | Π | / | f ( X ) | paths. Sin ce th ere are lg L + 1 buc kets, the normalized congestion of e in one of those buc kets is at least | Π | / ( | f ( X ) | · (lg L + 1)). Therefore, C ∗ ≥ | Π | / ( | f ( X ) | · (lg L + 1)). Since | f ( X ) | ≤ 2 | X | , w e obtain: | Π | ≤ 2 C ∗ · | X | · (lg L + 1) ≤ 4 C ∗ · | X | · lg n. (2) By Combining Equations 1 and 2, w e get: C ≤ 16 C ∗ · D ∗ · lg 2 n + 2 D ∗ lg n ≤ 18 C ∗ · D ∗ · lg 2 n. When C > D / 4, using Lemma 4.5 it is straigh tforwa r d to prov e that P oA = O (( C ∗ · D ∗ · lg 2 n ) / ( C ∗ + D ∗ )). If C ≤ D / 4, then u sing Lemma 4.3, we can pr o v e th at P oA = O (1) (the details can b e found in the app endix). Therefore, we obtain the main r esult: Theorem 4.6 (Price of anarc hy in sum-buc k et games) F or any sum-bucket game R it holds: P oA = O C ∗ · D ∗ C ∗ + D ∗ · lg 2 n ! . There is a sum-b uc ke t game R = ( N , G, P ) that sho ws that th e result of Theorem 4.6 is tight in non-trivial cases. All the pla yers hav e th e same source n o de u and destination no d e v . L et a = √ N (supp ose f or simplicit y th at a = √ N = ⌈ √ N ⌉ ). There is a path p of length a fr om u to v . There are a edge-disjoin t paths Q = { q 1 , . . . , q a } f r om u to v so that eac h path q i has length a and uses one edge of p . Eac h pla y er has t wo paths in h er strategy set: one is path p and the other is a p ath fr om Q . F urther, for eac h path q i there are a play ers that ha v e q i in their strategy s ets. Let p b e the routing where ev ery pla ye r chooses path p . Then, p is a Nash-routing with so cial cost a + N . Let p ′ b e th e r outing w here eve ry pla yer chooses the alternativ e path in Q . Then, p ′ is also a Nash-routing with so cial cost 2 a . Thus, P oA ≥ S C ( p ′ ) /S C ( p ) ≥ ( a + N ) / ( a + 1) = Ω ( √ N ) = Ω( √ n ). R ou tin g p ′ is an optim um routing with the smallest so cial cost C ∗ = C ( p ) = a and D ∗ = D ( p ) = a . Thus, from Theorem 4.6 , P oA = O (( C ∗ · D ∗ · lg 2 n ) / ( C ∗ + D ∗ )) = O ( a · lg 2 n ) = O ( √ n · log 2 n ). Hence, the pr ice of anarc hy has to b e within a log 2 n factor from the b oun d pro vided in Theorem 4.6. 5 Conclusions In this work w e pro vided the ﬁrst study (to our knowledge ) of b icriteria routin g games, where the pla ye rs attempt to sim ultaneously optimize t wo parameters: their p ath congestion and length. The motiv ation is the existence of eﬃcien t pac ke t scheduling algorithms wh ic h d eliv er the pac ket s in time prop ortional to the so cial cost. W e examined max games and sum games. Max games stabilize, but their price of anarch y is h igh. S um games do not stabilize, b u t they can giv e b etter price of anarc hy . W e then giv e th e approximate sum-b uc ke t games whic h alw a ys stabilize and p r eserv e the go o d prop erties of s um games. S urprisin gly , arbitrary sum -buc ket game equilibria provi d e go o d appro ximations to the original co ord inated routing problem. Sev eral op en problems remain to examine. W e studied tw o particular functions of the bicriteria, namely , the max and the s um fu n ctions. There are other fu nctions, for example a weigh ted s u m, th at could pro vide s im ilar or b ette r r esu lts. It would also b e in teresting to add additional parameters. The original C + D sum games do not stabilize in general. Ho wev er, there exist interesti n g instances whic h stabilize . F or example, it can b e easily sho wn that the games where the a v ailable paths ha ve equal lengths alw a ys stabilize. It would b e interesting to ﬁn d a general c haracterization of the game instances that stabilize. Another in teresting problem is to pro vide time eﬃcie nt algorithms for ﬁnd ing equilibria in our games. 10 References [1] E. Ansh elevic h, A. Dasgupta, Jon Kleinb erg, , E. T ardos, T. W exler, and Tim Roughgarden. The price of stabilit y for net wo r k design w ith fair cost allocation. In Pr o c. FOCS , 2004. [2] E. Anshelevic h , A. Da sgu p ta, E. T ardos, and T. W exler. Near optimal n et w ork design with selﬁsh agen ts. In P r o c. STOC , 2003. [3] P . Beren brink and C . Sc heideler. L o cally eﬃcien t on-line str ategie s for routing pac k ets along ﬁxed paths. In 10th A CM -SIAM Symp osium on Discr ete AL gorithms (SODA) , pages 112–1 21, 1999. [4] C ostas Bus c h and Malik Magdon-Ismail. A tomic routing ga m es on maxim um conge stion. In Pr o c e e dings of the 2nd Internat ional Confer enc e on Algorithmic Asp e cts in Informat ion and Management (AAIM) , p ages 79–9 1, Hong Kong, China, Jun e 2006. [5] G. Christo d ou lou and E. Koutsoupias. The price of anarc hy of ﬁnite congestion games. In Pr o c. STOC , 2005 . [6] J ose R. Correa, And reas S. Sch u lz, and Nicolas E. Stier Moses. Comp u tational complexit y , fairness, and th e pr ice of anarc h y of the maxim um latency prob lem. In Pr o c. 10th Conf. on Inte ger P r o gr amming and Combinatorial Optimization (IPCO) , 2004. [7] Rob ert Cypher , F riedh elm Mey er auf der Heide, Ch ristian Scheideler, and Berthold V¨ oc king. Univ ersal algorithms for store-and-forward and wo rm h ole routing. In In P r o c . of the 28th ACM Symp. on The ory of Computing , pages 356– 365, 1996. [8] A. Czuma j, P . Krysta, and B. V¨ oc king. S elﬁsh traﬃc allocation for serv er farms. In Pr o c. STOC , 2002. [9] A. Czuma j and B. V¨ oc king. Tight b ounds for worst-ca se equilibria. In P r o c . SODA , 2002. [10] D. F otakis, S. Kon togiannis, E. Koutsoupias, M. Ma vronicolas, and P . Spir akis. T he s tr ucture and complexit y of Nash equilibria for a selﬁsh routing game. In Pr o c. ICA LP , 2002. [11] D. F otakis, S. Konto giannis, and P . S p irakis. Selﬁsh un splittable ﬂows. In Pr o c. ICALP , 2004. [12] M. Garing, T. L ¨ uc king, M. Ma vronicolas, and B. Monien. Computing nash equilibria for sc hedulin g on restricted parallel links. In Pr o c. STOC , 2004. [13] M. Garing, T . L ¨ uc king, M. Ma vr onicolas, and B. Monien. The price of anarc hy for p olynomial so cial cost. In Pr o c. M FCS , 2004. [14] M. Garing, T. L ¨ uc king, M. Ma vronicolas, B. Monien, and M. Ro de. Nash equilibria in d iscrete routing games w ith con ve x latency fu nctions. In Pr o c. ICALP , 2004. [15] E. Koutsoupias, M. Ma vronicolas, and P . S pirakis. App ro ximate equilibria and ball fusion. In Pr o c. SIROCCO , 2002 . [16] E. Koutsoup ias and C. Papadimitriou. W orst-case equilibr ia. In Pr o c. ST ACS , 1999. 11 [17] F. T. Leight on, B. M. Magg s, and S. B. Rao. P ac ket routing and job-sc h eduling in O ( cong estion + dil ation ) steps . Combinatoric a , 14:167– 186, 199 4. [18] T om Leigh ton, Bruce Maggs, and Andrea W. Ric h a. F ast algorithms for ﬁnd ing O(congestion + dilation) p ac k et routing sc h edules. Combinatoric a , 19:375–401 , 1999. [19] Lavy Libman and Ariel Orda. A tomic resource sharing in nonco op erativ e net works. T ele c o- munic ation Systems , 17(4 ):385–409, 2001. [20] T. L ¨ u c king, M. Mavronico las, B. Monien, and M. Ro de. A new mo del for s elﬁsh routing. In Pr o c. ST ACS , 2004. [21] M. Ma vronicolas and P . S pirakis. Th e price of selﬁsh routing. In Pr o c. STOC , 2001. [22] D. Mond erer and L. S . Shap ely . Po tenti al games. Games and Ec onomic Behavior , 1996. [23] Rafail O s tro vsky and Y uv al Raban i. Univ ersal O (congestion+dilation+log 1+ ε N ) lo cal con trol pac k et switc hing algorithms. I n P r o c e e dings of the 29th Annual ACM Symp osium on the The ory of Computing , pages 644–653, New Y ork, Ma y 1997. [24] C. H. P apadimitriou. Algorithms, games and the internet. In Pr o c. STOC , p ages 749–753 , 2001. [25] Y uv al Rabani and ´ Ev a T ardos. Distributed pac k et switc hing in arbitrary net wo r ks. In Pr o- c e e dings of the Twenty-Eighth Annual A CM Symp osium on the The ory of Computing , pages 366–3 75, Philadelphia, P ennsylv ania, 22–24 May 1996 . [26] R. W. Rosenthal . A class of games p ossesing pure-strategy Nash equilibr ia. Internationa l Journal of Game The ory , 1973. [27] Tim R ou gh gard en . The maximum latency of selﬁsh routing. In Pr o c. SODA , 2004. [28] Tim R ou gh gard en . Selﬁsh routing with atomic p la y ers. I n P r o c . SODA , 2005. [29] Tim Roughgarden and ´ Ev a T ardos. Ho w b ad is selﬁsh routing. Journal of the ACM , 49(2):236 – 259, Marc h 2002. [30] Tim R ou gh gard en and ´ Ev a T ardos. Bounding the ineﬃciency of equilibr ia in non atomic con- gestion games. Games and Ec onomic Behavior , 47(2):38 9–403, 2004 . [31] S. S uri, C . D. T ´ oth, and Y. Zhou. Selﬁsh load balancing and atomic congestion games. In Pr o c. SP AA , 2004. 12 A Additional Pro ofs of Section 3.1 Lemma A.1 Minimum r outing p min achieves optimal so cial c ost, that i s, S C ( p min ) ≤ S C ( p ) , for any other r outing p 6 = p min . Pro of: Supp ose for con tradiction that there exists a routing p ≥ p min with S C ( p ) < S C ( p min ). Let S C ( p ) = max( C ( p ) , D ( p )) = k 1 , and S C ( p min ) = max( C ( p min ) , D ( p min )) = k 2 . Clearly , k 1 < k 2 . Th erefore, in the v ector M ( p ) = [ m 1 , . . . , m r ] it h olds that m k 1 6 = 0 and m k = 0 for k > k 1 . S imilarly , in the v ector M ( p min ) = [ b m 1 , . . . , b m r ] it h olds that b m k 2 6 = 0 and b m k = 0 for k > k 2 . T herefore, M ( p min ) > M ( p ), con tradicting the fact that p ≥ p min . B Additional Pro ofs of Section 4.1 In th e table b elo w we calculate the congestions, path lengths, and pla y er costs for eac h routing scenario of the game in Theorem 4.1. W e set the sp eciﬁc lengths as: | p 1 | = 10, | p ′ 1 | = 8, | p 2 | = 7, | p ′ 2 | = 10, | p 3 | = 7, and | p ′ 3 | = 10. F or a p la y er i ∈ { 1 , 2 , 3 } and routing p w e deﬁne the c omplementary r outing to b e the one where p la y er i c ho oses the alternativ e path. F or example, for pla yer 2 the complemen tary routing of [ p 1 , p 2 , p 3 ] is [ p 1 , p ′ 2 , p 3 ]. A play er is lo cally optimal in a routing if the complementary r outing d o es not give a low er cost for the pla yer. Using the table it is easy to determine wh ether a pla y er is lo cally optimal or n ot b y examining the resp ectiv e costs in the complementa r y routings. In this wa y , w e ﬁ nd th e non-lo cally pla y ers which are s h o wn in the righ tmost column of the table. routing C 1 D 1 pc 1 C 2 D 2 pc 2 C 3 D 3 pc 3 Non lo cally optimal pla yers [ p 1 , p 2 , p 3 ] 4 10 14 4 7 11 4 7 11 pla ye r 1 [ p ′ 1 , p 2 , p 3 ] 5 8 13 5 7 12 5 7 12 pla yer 2 [ p ′ 1 , p ′ 2 , p 3 ] 5 8 13 1 10 11 5 7 12 pla ye r 3 [ p ′ 1 , p ′ 2 , p ′ 3 ] 5 8 13 1 10 11 1 1 0 1 1 p la y er 1 [ p 1 , p ′ 2 , p ′ 3 ] 2 10 12 2 1 0 1 2 2 10 12 play er 2 [ p 1 , p 2 , p ′ 3 ] 3 10 13 4 7 11 2 1 0 1 2 p la y er 3 [ p 1 , p ′ 2 , p 3 ] 3 10 13 2 10 12 4 7 11 pla y er 2 [ p ′ 1 , p 2 , p ′ 3 ] 5 8 13 5 7 12 1 10 11 pla y er 2 C Additional Pro ofs of Section 4.2.1 Lemma C.1 If a g r e e dy move b y player i takes a r outing p to a new r outing p ′ , then p ′ < p . Pro of: Let p i and p ′ i denote the paths of i in routings p and p ′ , resp ectiv ely . Let pc i ( p ) = C i ( p ) + D i ( p ) = z 1 and pc i ( p ′ ) = C i ( p ′ ) + D i ( p ′ ) = z 2 . S ince play er i decreases its cost in p ′ , z 2 < z 1 . Consider now the vect ors of the r outings M ( p ) = [ m 1 , . . . , m r ] and M ( p ′ ) = [ m ′ 1 , . . . , m ′ r ]. W e will sho w that M ( p ) < M ( p ′ ). Let B ( p i ) = k 1 and B ( p ′ i ) = k 2 . Let V d enote the set of p la y ers w hose cost increases in p ′ with resp ect to their cost in p . Next, we show that for any j ∈ V it holds th at p c j ( p ′ ) ≤ c i ( p ′ ), whic h will help u s to pro ve the desired r esult. Let p j b e the path of j ∈ V . Let B ( p j ) = k 3 (note th at j do es not switc h paths an d buc ket s b et ween p and p ′ ). If k 3 6 = k 1 and k 3 6 = k 2 then the cost of j remains u naﬀected b et w een p and i p ′ . Th u s, either k 3 = k 1 or k 3 = k 2 . If k 3 = k 1 , then the cost of j can only decrease fr om p to p ′ , since path p i can no longe r aﬀect p ath p j . T herefore, it has to b e that k 3 = k 2 . Supp ose, for the sak e of contradict ion, that pc j ( p ′ ) > pc i ( p ′ ). Then, C j ( p ′ ) + D j ( p ′ ) > C i ( p ′ ) + D i ( p ′ ). Since b oth paths are in the s ame buc ket , D j ( p ′ ) = D i ( p ′ ), which implies that C j ( p ′ ) > C i ( p ′ ). Since j ∈ V , C j ( p ) < C j ( p ′ ). The increase in normalized congestion of p j in p ′ can only b e caused by p ′ i b ecause it sh ares an edge with p j with congestio n C j ( p ′ ). Ho wev er, this is imp ossible since it w ould im p ly that C i ( p ′ ) ≥ C j ( p ′ ). Th erefore, pc j ( p ′ ) ≤ pc i ( p ′ ). Consequent ly , in vect or p ′ all the en tries in p ositio ns j 2 + 1 , . . . , r d o not increase with r esp ect to p . F urth er, b ecause i s w itc hes p aths, m j 1 > m ′ j 2 . T h us, M ( p ) < M ( p ′ ), as n eeded. D Additional Pro ofs of Section 4.2.2 Lemma D.1 In N ash-r outing p , for every player i and e dge e it holds that | f ( e, i ) | ≥ 1 . Pro of: Supp ose that | f ( e, i ) | = 0. Then for ev ery edge e ′ ∈ p ∗ i it holds that C e ′ ,p ∗ i ( p ) < C e,p i ( p ) − D ∗ . Let C ′ = max e ′ ∈ p ∗ i C e ′ ,p ∗ i ( p ). If pla yer i was to choose path p ∗ i its cost would b e: pc ′ i ≤ C ′ + 1 + D p ∗ i < C e,p i ( p ) − D ∗ + 1 + D ∗ = C e,p i ( p ) + 1 ≤ C i ( p ) + D i ( p ) = pc i ( p ) . Thus, pc ′ i < pc i ( p ) whic h implies that p la y er i is not lo cally optimal in Nash-r ou tin g p , a con tradiction. Pro of of Theorem 4.6: Supp ose that p is the w orst Nash-routing with maxim u m so cia l cost. W e h a v e that P oA ≤ S C ( p ) /S C ( p ∗ ) ≤ ( C + D ) / ( C ∗ + D ∗ ) . W e examine t wo cases: • C > D / 4: In th is case D = O ( C ). F rom Lemma 4.5, C ≤ 18 C ∗ · D ∗ · lg 2 n . Therefore: P oA = O ( C / ( C ∗ + D ∗ )) = O (( C ∗ · D ∗ · lg 2 n ) / ( C ∗ + D ∗ )) . • C ≤ D / 4: Let p i ∈ p b e the path with maximum cost in p . Clearly , pc i ( p ) = C i + D i ≥ D . F urther, 0 ≤ C i ( p ) ≤ C ≤ D / 4. Th us, D i ≥ D − C i ≥ D − D / 4 = 3 D / 4. Sin ce C i ≥ 0 = C − C , Lemma 4.3 giv es | p i | ≤ | s i | + C + 1 ≤ D ∗ + D / 4 + 1. W e ha ve that D i / 2 < | p i | . T herefore, D i / 2 < D ∗ + D / 4 + 1, whic h giv es: 3 D / 8 < D ∗ + D / 4 + 1. Th u s, D < 8( D ∗ + 1) ≤ 16 D ∗ . In order wo r ds, D = O ( D ∗ ). Since, C = D , w e obtain: P oA = O ( D / ( C ∗ + D ∗ )) = O ( D ∗ / ( C ∗ + D ∗ )) = O (1) . By com bin ing the tw o ab ov e cases we obtain the desirable r esu lt. ii Bicriteria Optimization in R outing Games Costas Busc h Computer Science Department Louisiana State Univ ersit y 280 Coates Hall Baton Rouge, LA 70803 , USA busch@csc.lsu.e du Ra jgopal Kannan Computer Science Department Louisiana State Unive rsity 279 Coates Hall Baton Rouge, LA 7 0803, USA rk annan@csc.lsu.edu Abstract Two impo rtant metrics for measuring the qualit y of routing paths are the ma x im um edge co n- gestion C and ma ximum path length D . Here, we study bicr iteria in r outing games where each play er i selﬁshly se le cts a path that simultaneously minimizes its maxim um e dg e c o ngestion C i and pa th length D i . W e s tudy the stability and price of a narch y o f tw o bicr iteria games: • Max games , where the so cial cos t is max( C, D ) and the play er cost is max ( C i , D i ). W e prov e that ma x g ames are sta ble and c onv ergent under b es t- r esp onse dyna mics , and that the price of a na rch y is b ounded ab ov e by the maximum path length in the play er s’ strategy sets. W e also show that this b ound is tight in w o rst-case scenario s. • Sum games , wher e the so cial cost is C + D and the play er cost is C i + D i . F or sum games, we ﬁrst s how the negative res ult that there are game instances that hav e no Nas h- equilibria. Therefore, w e examine a n approximate game ca lled the sum-bucket game that is always conv ergent (and therefore sta ble). W e show that the pr ice of anarch y in sum-buck et games is bo unded ab ov e by C ∗ · D ∗ / ( C ∗ + D ∗ ) (with a p oly-lo g factor ), whe r e C ∗ and D ∗ are the optimal co ordina ted co ng estion and path leng th. Th us, the sum-buck et game has typically sup e rior price of ana rch y b ounds than the max game. In fact, when either C ∗ or D ∗ is small (e.g. constant) the so c ia l cost of the Nash-equilibria is very clos e to the co or dinated optimal C ∗ + D ∗ (within a p oly-log factor). W e also show that the pr ice of anarch y b ound is tight for case s where b oth C ∗ and D ∗ are lar ge. Regular presen tation for SP AA 1 In tro du c tion Routing is a fundament al task in comm unication net works. Routing algorithms provi d e paths for pack ets that will b e sen t o ve r th e n et w ork. There are t wo metrics that quan tify the qualit y of the paths return ed b y a routing algorithm: the congestion C , wh ich is th e maxim um num b er of paths that use any edge in the n et w ork, and the maximum path length D . Assuming there is a p ac k et for eac h p ath, a lo wer b ound on the delive r y time of the pac ket s is Ω(max( C, D )) (alternativ ely , Ω( C + D )). Actually , th er e exist pac k et sc h eduling algorithms that give n the paths, they deliv er the p ac k ets along the paths in time close to optimal O (max( C, D )) (alternativ ely , O ( C + D )) [ ? , ? , ? , ? , ? ]. Motiv ated by the selﬁsh b eha vior of en tities in communicatio n net works, we study routing games where eac h pac ket’ s path is con trolled indep enden tly by a selﬁsh pla y er. W e mo del games with N pla ye rs , where eac h pla yer h as to s elect a path fr om a source to a destination no de. The ob jectiv e of eac h p lay er i is to select a path that sim ultaneously minimizes tw o parameters: th e congestion C i , whic h is the maxim um n u mb er of paths that use any edge in pla y er i ’s path, and the path length D i . W e examine t wo kind s of games: max games , where the pla y er’s cost function is max( C i , D i ), and sum games , where the pla y er’s cost fu n ction is C i + D i . In eac h of these games, th e play er’s ob jectiv e is to selﬁsh ly minimize its cost in an unco ordinated manner. F rom the pla y er’s p oin t of view, th e m inimization of the sum or max cost functions are justiﬁed ob jectiv es, sin ce it is sho wn in [ ? ] that p la y er i ’s pack et can b e deliv ered in time ˜ O ( C i + D i ) (alternativ ely , ˜ O (max( C i , D i )). A natural pr oblem is to determine th e eﬀect of the p lay ers ’ selﬁshness on the w elfare of the wh ole comm unication net wo r k . In the max and sum games, the w elfare of the n et w ork is measur ed with the so cial cost functions max( C, D ) and C + D , resp ect ively . The c hoice of these so cial cost functions is appr opriate since th ey determine the total time needed to d eliv er the pack ets repr esen ted b y the pla ye rs . W e examine the consequ ence of the selﬁsh b eha vior in Nash equilibria, which are stable states of the game w here no pla y er can unilaterally impro ve her situation. The eﬀect of selﬁshness is quant iﬁed with the pric e of anar chy ( P oA ) [ ? , ? ], which exp resses ho w m u c h larger is the wo r st so cial cost in a Nash equilibrium compared to the so cial cost in the optimal co ordinated solution. W e study the existence of Nash equilibria and the p rice of anarc hy for max games and sum games, where we ﬁnd that these games pro du ce diﬀerent results with their o w n mer its. 1.1 Max Games First, w e examine max games (the so cial cost is max( C + D )). W e pro ve that ev ery max game has at least one Nash equilibrium. Th e equilibriu m can b e obtained b y b est resp onse d ynamics, where a pla y er greedily c hanges, whenever p ossible, the curr en t path to an alternativ e path with lo wer cost. With b est resp onse mo ves the game ev en tually conv erges to a Nash-equilibrium. W e sh ow that the optimal coord inated solution is a Nash-equilibr iu m to o. Th us, max games games hav e v ery go o d Nash equilibr ia. Th is observ ation is quanti ﬁ ed in terms of the pric e of stability ( P oS ) [ ? , ? ] whic h expr esses how muc h larger is the b est so cial cost in a Nash equilibriu m with resp ect to the so cial cost in the optimal co ord in ated solution. Therefore in max games it holds that P oS = 1. W e then examine the worst Nash equilibr ia. W e b ound th e price of anarc hy ( P oA ) in max games w ith resp ect to the maxim um allo wable path length L for the pla yers in the n et w ork, and the num b er of no d es n in the graph : P oA = O ( L + log n ) . 1 W e prov e that this b ound is worst case optimal (within add itiv e terms). S p eciﬁcally , we provide an example game in a ring net wo rk where the optimal co ordin ated so cial cost is 1, while there is a Nash equilibrium w ith cost O ( L ) = O ( n ). 1.2 Sum Games W e contin ue with examining sum games (the so cial cost is C + D ). Intuiti vely , sum games hav e the p oten tial to giv e b ette r price of anarc hy th an the max games b ecause b oth parameters ( C and D ) aﬀect th e c hoices at all the time, eve n w hen one parameter is larger than the other. F or example, the ring game that w e mentioned ab o ve has price of anarch y equal to 1 in the sum game. Ho w eve r , w e prov e a limitation of sum games: not all sum games ha v e Nash equilibria; there exist instances of sum games w ith a small n u m b er of pla y ers that do not hav e Nash equilibria at all. This limitation directed us to wards exploring alternativ e games w h ic h are stable (hav e Nash equilibria) and hav e similar c h aracteristics with the original sum games. W e foun d such a game v ariation that w e call sum-buck e t game . In su m-buck et games the pla y ers are d ivided into log n classes, called buck ets , according to the pac ke t paths that they choose. Buc ke t k holds the paths of p la y ers with length in range [2 k , 2 k − 1 ). S upp ose that pla ye r i ’s path is in b uc ke t k . The normalize d c ongestion of pla yer i , denoted C i , is measured with r esp ect to th e p aths that b elong to buc ke t k . The normalize d length of play er i ’s path is D i = 2 k − 1 − 1, (wh ich is a facto r 2 appro ximation of the original length). Pla yer i ’s cost function is C i + D i . Thus, in su m-buc ket games only play ers in the same buc ket comp ete with eac h other, while pla y ers in diﬀerent buc ket s do not interfere. The normalize d so cial c ost function is deﬁn ed to b e C + D , wh ere C is the maxim um normalized congestion in an y buc ket, and D is the m axim um n ormalized depth of all paths. W e ﬁrst sho w that sum-buck et games alw a ys hav e Nash equilibr ia, whic h can b e obtained with b est r esp onse dynamics. W e then examine the qualit y of the Nash equilibria. F or ev ery game there is a corresp onding co ordinated buck et r outing problem. W e can b ound the price of anarch y P oA w ith resp ect to the normalized optimal congestion C ∗ and normalized p ath length D ∗ in the optimal co ordinated solution. W e obtain: P oA = O C ∗ · D ∗ C ∗ + D ∗ · log 2 n ! Therefore, w h en either of C ∗ or D ∗ is small (e.g. a constant), the Nash equilibriu m p ro vides a v ery go o d appr o ximation (within a p oly-log factor from optimal) to the u nco ordinated routing problem. In suc h s cenarios, the pr ice of selﬁsh ness is sm all. How ev er, when b oth C ∗ or D ∗ are sim u ltaneously large, the appr o ximation b ecomes worse (though still typical ly lo wer th an th e P oA of the max game since the P oA is b ounded by the smaller of C ∗ or D ∗ , wh er e D ∗ < 2 L ). Nev erth eless, even in these scenarios the PoA b ound is tight in certain games instances. Sum-bu c k et games are in teresting v ariations of sum games b ecause they are stable, and they can b e used to appr o ximate solutions f or the C + D so cial cost. F or any s u m-buck et game, there is a corresp onding “original” co ordinated rou tin g problem where the ob jectiv e is to m inimize the so cial cost C + D without usin g b uc ke ts. It holds th at C ≤ C ≤ C · log n and D ≤ D ≤ 2 D ; th us, C + D = O ( C · log n + D ). In other w ords, the normalized so cial cost can b e used as an appro ximation for the “origi n al” so cial cost. Let P oA ′ denote ho w m uch larger is the w orst equilibrium of a s u m-buck et game compared to the optimal solution of the co ordin ated original problem (with resp ect to the so cial cost C + D ). It holds that P oA ′ ≤ P oA · log n . Consequently , 2 the observ ations that w e made ab o ve for the P oA in sum-b uc ke t games apply also with resp ect to the original r outing problem. F or example, wh en one of C ∗ or D ∗ is small (e.g. a constan t), then the Nash equ ilibrium of the sum -buc ket game pro vides a v ery goo d approxi mation (within a p oly log factor fr om optimal) to th e co ord inated original r outing problem. 1.3 Related W ork Routing games (on congestion) were introdu ced and stu died in [ ? , ? ]. The notion of price of anarc hy was introd uced in [ ? ]. S ince then, many r outing game mo d els h av e b een studied w hic h are distinguished by the top olog y of the netw ork, cost functions, t yp e of traﬃc (atomic or splittable), nature of strategy sets, and kind of equilibria (pu re or mixed). S p eciﬁcally , pu r e equilibria with atomic ﬂow h a v e b een stud ied in [ ? , ? , ? , ? , ? ] (our work ﬁts into th is category), and with s plittable ﬂo w in [ ? , ? , ? , ? ]. Mixed equilibria with atomic ﬂo w ha ve b een studied in [ ? , ? , ? , ? , ? , ? , ? , ? , ? , ? , ? ], and with splittable ﬂow in [ ? , ? ]. T o our kno wledge there is no previous work that considers routing games that optimize tw o criteria sim u ltaneously . Most of th e w ork in the literature uses a single cost metric whic h is related to the congestion. A common metric f or the p la y er cost is the sum of the congestions on all the edges of the pla y er’s path (w e denote this kind of play er cost as p c ′ ) and th e resp ectiv e so cial cost is the cost of the worst pla ye r ’s path (w e denote this so cial cost as S C ′ ) [ ? , ? , ? , ? , ? , ? ]. Ho we ver, as w e discussed b efore, in pac ket sc heduling alg orithm s , the pc ′ or S C ′ do not go v ern the p ac k et dela ys; m ax ( C i , D i ) or C i + D i go v ern the pack et d ela y . Other combinations of pla yer costs and so cia l cost s h a v e b een studied in the literature: p la y er cost pc ′ and so cial cost C has b een studied in [ ? , ? , ? , ? , ? , ? , ? , ? , ? , ? , ? , ? ]; pla y er cost C i and so cial cost C has b een studied in [ ? ]; other v ariations ha v e b een stud ied in [ ? , ? , ? , ? ]. T h e v ast ma jorit y of th e w ork on routing games has b een p erformed for parallel link netw orks, with only a few exceptions on general net work top ologies [ ? , ? , ? , ? ]. Our work is closer to [ ? ]. W e extended some results pr esen ted in [ ? ] to app ly to bicriteria, instead of the single criterium of congestion, pla y er cost C i and so cial cost C , that w as u sed in [ ? ]. Sp eciﬁca lly , the particular tec hniqu es that we use to prov e existence of Nash equilibria w ith b est resp onse d ynamics, and also to pr o v e u p p er b ounds on the price of anarch y , were originally in tro d uced in [ ? ]. Here, we mo diﬁed and extended appropr iately these tec hniques in a non-trivial w a y to apply to our new cost functions. Outline of Pa p er W e pro ceed as follo ws. In Section 2 w e giv e basic deﬁnitions. W e study max games in Section 3 and s um games in Section 4. W e ﬁn ish with our conclusions in Section 5. Due to sp ace limitations some pro ofs hav e mov ed to the ap p endix. 2 Deﬁnitions An ins tance of a r outing game is a tuple R = ( N , G, P ), where N = { 1 , 2 , . . . , N } are the pla yers, G = ( V , E ) is a graph with n o des V and edges E , and the graph has paths P = S i ∈ N P i , where P i is a collectio n of a v ailable paths in G for play er i . Eac h p ath in P i is a path in G that has the same source u i ∈ V and destinatio n v i ∈ V ; eac h path in P i is a pur e str ate gy a v ailable to play er i . A pur e str ate gy pr oﬁle p = [ p 1 , p 2 , · · · , p N ] is a collection of pure strategies (paths), one for eac h 3 pla ye r, where p i ∈ P i . W e refer to a pure strategy pr oﬁle as a r outing . On a ﬁnite n et w ork, a routing game is n ecessarily a ﬁn ite game. F or an y routing p and an y edge e ∈ E , th e e dge-c ongestion C e ( p ) is the num b er of paths in p that use edge e . F or an y path p , th e p ath-c ongestion C p ( p ) is the maxim um edge congestion o v er all edges in p , C p ( p ) = max e ∈ p C e ( p ). W e will u s e the notation C i ( p ) = C p i ( p ), f or any user i . T he network c ongestion is the maxim u m ed ge-conge stion o ve r all edges in E , that is, C ( p ) = max e ∈ E C e ( p ). W e denote the length (num b er of edges) of any path p as | p | . F or an y user i , we will also u se the notation D p i ( p ) or D i ( p ) to d enote the length | p i | . The longest p ath leng th in P is denoted L ( P ) = max p ∈P | p | . W e will d enote by D ( p ) the maxim um path length in routing p , that is D ( p ) = m ax p ∈ p | p | . When the con text is clear, we will d rop th e dep enden ce on p and R and use the n otation C e , C p , C i , C, L, D p , D i , D . F or game R and routing p , the so cial c ost (or glob al c ost ) is a function of routing p , and it is denoted S C ( p ). The player or lo c al c ost is also a f unction on p d enoted p c i ( p ). W e use the standard notation p − i to refer to the collect ion of paths { p 1 , · · · , p i − 1 , p i +1 , · · · , p N } , and ( p i ; p − i ) as an alternativ e notation for p whic h emphasizes the dep endence on p i . Pla yer i is lo c al ly optimal in routing p if pc i ( p ) ≤ pc i ( p ′ i ; p − i ) for all paths p ′ i ∈ P i . A routing p is in a Nash Equilibriu m (w e sa y p is a Nash-r outing ) if ev ery play er is lo cally optimal. Nash-routings quant ify the notion of a stable selﬁsh outcome. A routing p ∗ is an optimal pur e strategy p roﬁle if it has minim um attainable so cial cost: for an y other p ure strategy pr oﬁle p , S C ( p ∗ ) ≤ S C ( p ). W e quantify the qualit y of the Nash-routings by the pric e of anar chy ( P oA ) (sometimes r eferred to as the co ordination ratio) and the pric e of stability ( P oS ). Let P d enote the s et of distinct Nash- routings, and let S C ∗ denote the so cial cost of an optimal routing p ∗ . T hen, P oS = inf p ∈ P S C ( p ) S C ∗ , P oA = sup p ∈ P S C ( p ) S C ∗ . 3 Max Games Let R = ( N , G, P ) a routing game suc h that for an y routing p the so cial cost f unction is S C ( p ) = max( C ( p ) , D ( p )), and the play er cost f unction p c i ( p ) = max( C i ( p ) , D i ( p )). W e refer to suc h routing games as max games . First, we sho w that max games hav e Nash-routings and the price of stabilit y is 1. Then, w e b ound the p rice of anarc hy . 3.1 Existence of Nash-routings in Max Games W e sho w th at max games ha v e Nash-routings. W e pro ve this result by ﬁrst giving a totaly order for the rou tin gs using a form of lexico graph ic order in g. Then w e sho w th at an y greedy mo ve of a p la y er can only giv e a new routin g with smaller order. Thus, the greedy mo ve s w ill con v erge either to the smallest r outing or to a routing where no pla ye r can improv e furth er. In either case, a Nash-routing will b e reac hed. Let R = ( N , G, P ) b e a max r ou tin g game. Let r = max( N , L ). F or any routing p w e deﬁn e the r outing ve ctor M ( p ) = [ m 1 ( p ) , . . . , m r ( p )], wh ere m i ( p ) = a i ( p ) + b i ( p ), and a i ( p ) is the num b er of paths with congestion i , and b i ( p ) is the n umb er of paths with length i . Note that if S C ( p ) = k then m k 6 = 0 and m ′ k = 0 f or all k ′ > k . W e d eﬁne a total order on the routings as follo w s. Let p and p ′ b e t wo routings, with M ( p ) = [ m 1 , . . . , m r ], and M ( p ′ ) = [ m ′ 1 , . . . , m ′ r ]. W e sa y that M ( p ) = M ( p ′ ) if m i = m ′ i for all 1 ≤ i ≤ r . 4 W e say that M ( p ) < M ( p ′ ) if th ere is a j , 1 ≤ j ≤ r , such that m k = m ′ k for all k > j , and m j < m ′ j . W e ord er the p and p ′ according to th e order of their resp ectiv e v ectors, that is p ≤ p ′ if and only if M ( p ) ≤ M ( p ′ ). Note that for any t wo p and p ′ it either holds that p = p ′ or p < p ′ . That is, the routings are totally ordered. Consider an arbitrary routing p . If p is not a Nash-routin g, there is at least one u ser i w hic h is not lo cally optimal. Then a gr e e dy move is av ailable to play er i in wh ic h the play er can obtain lo w er cost b y changing the path from p i to some other path p ′ i with lo wer cost. In other words, the greedy mo v e tak es the original routing p = ( p i ; p − i ) to a routing p ′ = ( p ′ i ; p ′ − i ) with impro ved pla ye r cost pc i ( p ′ ) < pc i ( p ), su c h that p i is replaced by p ′ i and th e remaining paths sta y the same ( p − i = p ′ − i ). W e sho w no w that an y greedy mo ve giv es a smaller ord er routing: Lemma 3.1 If a gr e e dy move b y any player takes a r outing p to a new r outing p ′ , then p ′ < p . Pro of: Let pc i ( p ) = max( C i ( p ) , D i ( p )) = k max 1 , and min( C i ( p ) , D i ( p )) = k min 1 (clearly , k max 1 ≥ k min 1 ). Let also pc i ( p ′ ) = max( C i ( p ′ ) , D i ( p ′ )) = k max 2 , and min( C i ( p ′ ) , D i ( p ′ )) = k min 2 (clearly , k max 2 ≥ k min 2 ). Since pla y er i can decrease its cost in p ′ , k max 2 < k max 1 . Consider no w the vec tors M ( p ) = [ m 1 , . . . , m r ] and M ( p ′ ) = [ m ′ 1 , . . . , m ′ r ]. These tw o ve ctors are the same except p ossibly for en tries k max 1 , k min 1 , k max 2 , k min 2 , whic h corresp ond to the p ositions that are aﬀected by p aths p i and p ′ i . It holds that m k max 1 > m ′ k max 1 , since when the path switches to p ′ i , m k max 1 = a k max 1 + b k max 1 decreases by at least one b ecause either a k max 1 decreases by one (if the new path h as lo wer congestion) or b k max 1 decreases by on e (if the new p ath has low er length). Since k max 2 < k max 1 , M ( p ) > M ( p ′ ) implying that p > p ′ . Since th er e are only a ﬁ nite num b er of routings, Lemma 3.1 implies that starting from arb itrary initial state, ev ery b est resp onse d y n amic con v erges in a ﬁn ite time to a Nash-routing, where ev ery pla ye r is lo cally optimal. Since the routings are tota lly ordered, there is a routing p min whic h is the minimum, that is, f or all routin gs p , p min ≤ p . Clearly , the minimum routing is also a Nash- routing. The minim um routing p min ac hiev es also optimal social cost, since if there w as another routing p ′ with lo w er so cial cost, then it can b e easily sh o wn th at p ′ < p min , wh ic h is con tradiction. Th u s, the price of s tabilit y is 1. Therefore, we h a v e the follo wing result: Theorem 3.2 (St abilit y of max games) F or any max g ame R , every b est r esp onse dynamic c onver ges to a Nash-r outing, and the pric e of stability is P oS = 1 . 3.2 Price of Anarch y in Max Games W e b ound the price of anarc hy in max games. Cons id er a max routing game R = ( N , G, P ), where G h as n no des. Theorem 3.2 imp lies th at there is at least one Nash-routing. Consider a Nash- routing p . Denote C = C ( p ) and D = D ( p ). Let p ∗ b e the optimum (coordin ated) routing with minim u m so cial cost. Denote C ∗ = C ( p ∗ ) and D ∗ = D ( p ∗ ). Note that eac h pa yer i ∈ N h as a p ath p i ∈ p and a corresp ond ing “optimal” path p ∗ i ∈ p ∗ from the play er’s source to th e destination. F or eac h edge e ∈ G , d enote Π e ( p ) the set of pla y ers wh ose paths in routing p use edge e . W e deﬁne H to b e a set that con tains all edges e ∈ G with congestion C e ( p ) ≥ D + 2. Consider an edge e ∈ H . L et i ∈ Π e ( p ) b e a pla y er whose p ath p i in routing p uses edge e . W e d eﬁne f ( e, i ) to b e a set th at con tains all edges e ′ ∈ p ∗ i with C e ′ ( p ) ≥ C e ( p ) − 1. It holds that | f ( e, i ) | ≥ 1, since in routing p play er i p r efers path p i instead of p ∗ i b ecause th ere is at least one edge e ′ ∈ p ∗ i 5 with C e ′ ( p ) ≥ C e ( p ) − 1 > D . Let f ( e ) = ∪ i ∈ Π e ( p ) f ( e, i ). F or any set of edges X ⊂ H , w e deﬁ ne f ( X ) = S e ∈ X f ( e ). Lemma 3.3 L et Z b e the set that c ontains al l e dges e with c ongestion C e ( p ) ≥ C − 2 log n . If C ≥ D + 2 lg n + 2 , then ther e is a set of e dges X ⊆ Z suc h that | f ( X ) | ≤ 2 | X | . Pro of: W e r ecursiv ely construct a sets of edges E 0 , . . . , E 2 lg n , s uc h that E i = E i − 1 ∪ f ( E i − 1 ), and set E 0 con tains all the edges e with congesti on C e ( p ) = C . F rom the construction of those sets it holds that for an y e ∈ E j , C e ( p ) ≥ C − j , where 0 ≤ j ≤ 2 lg n . Th us, E j ⊆ Z , for all 0 ≤ j ≤ 2 lg n . (Note that Z ⊆ H .) W e can show th at there is a j , 0 ≤ j ≤ 2 lg n , su c h th at | f ( E j ) | ≤ 2 | E j | . Supp ose for contra- diction th at su c h a j do es not exist. Thus f or all j , 0 ≤ j ≤ 2 lg n , it holds that | f ( E j ) | > 2 | E j | . In this case, it it straigh tforwa r d to sho w that | E k | > 2 | E k − 1 | , for an y 1 ≤ k ≤ 2 lg n . S ince | E 0 | ≥ 1, it holds that | E 2 lg n | > 2 2 lg n = n 2 . Ho we ver, this is a con tradiction, since the num b er of edges in G do not exceed n 2 . Lemma 3.4 If C ≥ D + 2 lg n + 2 , then C < 2 LC ∗ + 2 lg n . Pro of: F rom Lemma 3.3, there is a set of edges X ⊂ Z w ith | f ( X ) | ≤ 2 | X | . F or eac h e ∈ Z it holds th at C e ( p ) ≥ C − 2 lg n . Let M = P e ∈ X C e ( p ) ≥ | X | ( C − 2 lg n ), where M denotes the total utilization of the edges in X b y th e p aths of the pla yers in Π. Let Π = S e ∈ X Π e ( p ), that is, Π is the set of pla ye r s whic h in r outing p their paths use edges in X . By constru ction, the congestion in routing p in eac h of the edges of X is caused only b y the p la y ers in Π. Sin ce path lengths are at most L , eac h pla y er in Π can use at most L edges in X . Hence, M ≤ L · | Π | . Cons equen tly , | X | ( C − 2 lg n ) ≤ L · | Π | , wh ic h giv es: C ≤ ( L · | Π | ) / | X | + 2 lg n . By the d eﬁnition of f ( X ), in the optimal routing p ∗ eac h user in Π has to use at least one edge in f ( X ). Thus, edges in th e optimal routing p ∗ , the edges in f ( X ) are used at least Π times. Th u s, there is some ed ge e ∈ f ( X ) with C e ( p ∗ ) ≥ | Π | / | f ( X ) | . Therefore, C ∗ ≥ | Π | / | f ( X ) | . Since | f ( X ) | ≤ 2 | X | , we obtain | Π | ≤ 2 C ∗ · | X | . Therefore: C ≤ 2 LC ∗ + 2 lg n . Theorem 3.5 (Price of anarc hy in max games) F or any max game R it holds that P oA = O ( L + log n ) . Pro of: Supp ose that p is the worst Nash-r outing with maximum so cial cost. W e ha ve P oA = S C ( p ) /S C ( p ∗ ). If C ≥ D + 2 lg n + 2, then S C ( p ) = C . F rom Theorem 3.4, P oA ≤ C / max( C ∗ , D ∗ ) ≤ (2 LC ∗ + 2 lg n ) /C ∗ ≤ 2 L + 2 lg n . If C < D + 2 lg n + 2, then S C ( p ) < D + 2 lg n + 2; thus P oA ≤ L + 2 lg n + 2. Hence, in b oth cases P oA = O ( L + log n ). There is a max game th at shows that the result of Theorem 3.5 is tigh t in the wo r st case. Consider a ring net wo r k with n no des and n edges. Giv e the same orienta tion to the edges, so that eac h edge has one left n o de and one right no d e. F or eac h edge e i , there is a corresp onding play er i whose source is the left n o de and the d estination is th e right no de of the edge. T he strategy s et of eac h pla ye r h as t w o paths: p ath p i whic h is only th e edge e i , and path p ′ i whic h go es around the ring. Note that p = [ p 1 , p 2 , . . . , p n ] is a Nash-routing with so cial cost 1. Ho wev er, p ′ = [ p 1 , p 2 , . . . , p n ] is also a Nash rou tin g with so cial cost n − 1. Thus, the price of anarch y is O ( n ) = O ( L ). 6 4 Sum Games Let R = ( N , G, P ) b e a routing game suc h that for any routing p the so cial cost function is S C ( p ) = C ( p ) + D ( p ), and the pla y er cost function pc i ( p ) = C i ( p ) + D i ( p ). W e refer to suc h routing games as sum games . W e ﬁrst s h o w th at such games ha ve instances without Nash-routings. Then w e describ e a v ariation of sum games, that w e call sum-bu ck et games, w hic h are stable and their equilibria h a v e go o d p rop erties. 4.1 A Sum Game w it hout Nash-routings Theorem 4.1 Ther e is sum game instanc e R = ( N , G, P ) that has no Nash-r outing. Pro of: The graph G is d ep icted in the ﬁgure b elo w. There are sev en pla yers, namely , N = { 1 , . . . , 7 } . Pla ye r s 1, 2, and 3, hav e r esp ectiv e str ategy sets P 1 = { p 1 , p ′ 1 } , P 2 = { p 2 , p ′ 2 } , and P 3 = { p 3 , p ′ 3 } . I n the ﬁgure for pla ye r i = 1 , . . . , 3 the resp ectiv e source and d estination no des are u i and v i . There are six critical edges denoted e 1 , . . . , e 6 that the paths use and which are sho w n in the ﬁgure as straight horizon tal lines. Th ese edges ma y ha ve congestion larger than 1. The squiggly part of the p aths are assumed to ha v e congest ion 1 and their length and their lengths are c hosen so that th e f ollo w ing relations hold: | p ′ 1 | = | p 1 | − 2, | p ′ 2 | = | p 2 | + 3, an d | p ′ 3 | = | p 3 | + 3 . p 1 p 1 p ′ 1 p ′ 1 p 2 p 2 p 3 p 3 p ′ 3 p ′ 3 p ′ 2 p ′ 2 p 1 p ′ 3 p 1 p 2 p 3 p 1 p ′ 2 p ′ 1 p 2 p ′ 1 p 3 u 1 u 2 u 3 e 1 e 2 e 3 e 4 (+1) v 1 e 6 e 5 p ′ 1 (+3) (+3) (+4) v 2 v 3 p 2 p 3 Pla y ers 4 to 7 are “p assive” in the sense that they ha v e only one p ath in their strateg y sets, and their sole purp ose is to create additional congestion on edges e 3 , e 4 , e 5 , e 6 (the paths of these pla ye rs are n ot shown explicitly in the ﬁgure). In particular, the passiv e p la y ers cause additional congestion 1 to edge e 3 , additional congestion 3 to e 4 and e 5 , and additional congestion 4 to e 6 . The additional congestion is depicted in the ﬁgure in side a parent h esis und er eac h edge. Since the only “activ e” pla ye rs are 1, 2, and 3, and eac h pla yer has t wo path c hoices, there are eigh t p ossible d iﬀeren t routings. W e examine eac h r outing and pro v e that it is not a Nash-r outing. W e use th e v ector [ p 1 , p 2 , p 3 ] to denote a routing wh ere the i th p ositio n of th e vecto r conta ins the path choice of user i . By setting explicit v alues to the p ath lengths, and computing the p la y er costs in eac h rou tin g, w e ﬁnd that: pla y er 1 is not lo cally optimal in r outings [ p 1 , p 2 , p 3 ] and [ p ′ 1 , p ′ 2 , p ′ 3 ]; pla ye r 2 is not lo cally optimal in routings [ p ′ 1 , p 2 , p 3 ], [ p 1 , p ′ 2 , p ′ 3 ], [ p 1 , p ′ 2 , p 3 ], and [ p ′ 1 , p 2 , p ′ 3 ]; and pla ye r 3 is not lo cally optimal in routings [ p ′ 1 , p ′ 2 , p 3 ] and [ p 1 , p 2 , p ′ 3 ]. 4.2 Sum-buc ket Games Here we describ e sum-bucket games , whic h are v ariation of sum games that are stable and their equilibria ha ve go o d prop erties. Let R = ( N , G, P ) d enote a sum-buc ket r ou tin g game. Th e paths 7 in P are d ivided into buckets B 0 , . . . , B lg L so that B k is a set that con tains all paths wh ose lengths are in r ange [2 k , 2 k +1 ). (W e use lg L ins tead of ⌈ lg L ⌉ to a vo id notatio nal clutter.) F or any p ath p ∈ P , let B ( p ) denote the in d ex of the b uc ket that p b elongs to; namely , if p ∈ B k , then B ( p ) = k . Consider no w a routing p . W e deﬁn e the normalize d length of path p ∈ p as D p ( p ) = 2 B ( p )+1 − 1 (whic h is the maximum p ossib le path length in buck et B ( p )). Note that all the paths in the same buc ket ha ve the same normalized length. F or any path p we deﬁ n e th e normalize d c ongestion C p ( p ) to b e the maxim um congestion on an y of th e edges of path p whic h is caused by the paths of r outing p whic h b elong to b uc ket B ( p ). Pla y er i ’s cost is pc i ( p ) = C i ( p ) + D i ( p ). Th e pla ye r is allo we d to switc h from one bu c k et to another. The normalized congesti on of p is C ( p ) = max p i ∈ p C i ( p ), and the norm alized length of p is D ( p ) = max p i ∈ p D i ( p ). Th e so cial cost of game R is S C ( p ) = C ( p ) + D ( p ). Belo w, w e s h o w that sum-b u c ke t games are stable an d then we b ound the price of anarch y . 4.2.1 Existence of Nash-routings in Sum-buck et Games Here we sho w that sum-bu c ke t games hav e Nash-routings. W e use the same tec hnique as in S ec- tion 3.1, where w e order the routings and pro ve that greedy mov es giv e smaller order r outings. Let R = ( N , G, P ) b e a sum-buck et routing game. L et r = N + 2 L − 1 (this is the maxim um p ossible pla ye r cost). F or any r outing p w e deﬁne the r outing ve ctor M ( p ) = [ m 1 ( p ) , . . . , m r ( p )] such that m j ( p ) is th e n u mb er of paths in p with cost j . W e deﬁn e a total ord er on the routings, with r esp ect to their vecto rs, in exactly the same wa y as wee did for the max games in Section 3.1. Using similar tec hniques as in the max games, w e can prov e that if a greedy mov e b y pla y er i tak es a routing p to a new routing p ′ , then p ′ < p . Since there are only a ﬁnite num b er of routings, eve r y b est resp onse dynamic con v erges in a ﬁnite time to a Nash-routing. Th erefore, w e get: Theorem 4.2 (St abilit y of sum-buc k et games) F or any sum-buc k et g ame R , every b est r e- sp onse dynamic c onver ge s to a N ash-r outing. 4.2.2 Price of Anarch y in Sum-buck et Games F rom Theorem 4.2, eve r y sum-buck et game has at least one Nash-routing. Here, we b ound the pr ice of anarc h y . Cons ider a sum-bu c k et routing game R = ( N , G, P ), where G h as n no d es. Consider a Nash-routing p . Denote C = C ( p ) and D = D ( p ). Let p ∗ b e the optimum (co ordin ated) routing with min im um so cial cost. Denote C ∗ = C ( p ∗ ) and D ∗ = D ( p ∗ ). Note that eac h pay er i ∈ N has a path in p i ∈ p an d a corresp ondin g “optimal” path p ∗ i ∈ p ∗ from the pla yer’s source to th e destination no d e. F or an y pla yer i , let s i denote the shortest path in P wh ic h connects the sour ce and destinations n o des of i . W e no w relate the paths lengths with the congestion: Lemma 4.3 In Nash-r outing p , for any player i with C i ≥ C − x , wher e x ≥ 0 , it holds that | p i | ≤ | s i | + x + 1 . Pro of: Supp ose for the sake of con tradiction that there is a play er i with | p i | > | s i | + x + 1. Then, pc i = C i + D i ≥ C − x + D i ≥ C − x + | p i | > C − x + | s i | + x + 1 = C + | s i | + 1 . Clearly , | s i | ≤ 2 B ( s i )+1 − 1. I f user i was to switch to p ath s i its cost wo u ld b e p c ′ i ≤ C + 1 + 2 B ( s i )+1 − 1, since the normalized length of s i is 2 B ( s i )+1 − 1, and s i has normalized congestion at most C b efore pla ye r i switc hes, and the normalized congestion of path s i increases to at most C + 1 after p la y er i switc hes to it. Therefore, pc ′ i ≤ C + 1 + | s i | < pc i . Thus, in p p la y er i w ould not b e optimal, which is a con tradiction, sin ce p is a Nash-routing. Therefore, | p i | ≤ | s i | + x + 1, as needed. 8 F or eac h edge e ∈ G let Π e ( p ) denote the set of pla y ers wh ose paths in rou tin g p u s e edge e . Let C e,p i ( p ) denote the num b er of pac k ets th at use edge e in p and are in the same b uc ke t as p i . Let C e ( p ) = max p i C e,p i ( p ) d enote the normalized congestion of edge e . F or any edge e , w e deﬁne f ( e, i ) to b e a set that con tains all edges e ′ ∈ p ∗ i with C e ′ ,p ∗ i ( p ) ≥ C e,p i ( p ) − D ∗ . L et f ( e ) = ∪ i ∈ Π e ( p ) f ( e, i ), and f or an y set of edges X , f ( X ) = S e ∈ X f ( e ). It can b e shown that in Nash-routing p it holds | f ( e, i ) | ≥ 1. Lemma 4.4 L et Z b e the set that c ontains al l e dges e with c ongestion C e ( p ) ≥ C − 2 D ∗ · lg n . Ther e is a set of e dges X ⊆ Z with | f ( X ) | ≤ 2 | X | . Pro of: W e r ecursiv ely construct a sets of edges E 0 , . . . , E 2 lg n , s uc h that E i = E i − 1 ∪ f ( E i − 1 ), and set E 0 con tains all the edges e with congestion C e ( p ) = C . W e can sho w that th ere is a j , 0 ≤ j ≤ 2 lg n , su ch that | f ( E j ) | ≤ 2 | E j | . Sup p ose for cont rad iction that suc h a j do es not exist. Th u s for all j , 0 ≤ j ≤ 2 lg n , it holds that | f ( E j ) | > 2 | E j | . In this case, it it straigh tforw ard to sho w that | E k | > 2 | E k − 1 | , for an y 1 ≤ k ≤ 2 lg n . S in ce | E 0 | ≥ 1, it holds that | E 2 lg n | > 2 2 lg n = n 2 . Ho w ever, this is a con tradiction, sin ce the num b er of edges in G cannot exceed n 2 . Th us, there is a j with | f ( E j ) | > 2 | E j | . W e will set X = E j . It only remains to sho w that X ⊆ Z . It suﬃces to sh o w that for an y E k and e ∈ E k , C e ( p ) ≥ C − k D ∗ , where 0 ≤ k ≤ 2 lg n . W e pro ve this b y induction on k . F or k = 0 w e hav e that ev ery edge in e has C e ( p ) = C = C − 0 · D ∗ , th us the claim trivially holds. F or the induction hyp othesis, supp ose th at the claim holds for an y k = t < 2 log n . In the induction step w e will prov e that the claim holds also for k = t + 1. W e hav e that E t +1 = E t ∪ f ( E t ). By induction hyp othesis, for an y e ∈ E t , C e ( p ) ≥ C − tD ∗ (th us, f or any e ∈ E t it holds that C e ( p ) ≥ C − ( t + 1) D ∗ )). Thus, for any e ∈ E k there is at least one path p i with C e,p i ( p ) ≥ C − tD ∗ . By deﬁnition of f ( e, i ), ev ery edge e ∈ f ( e, i ) has the p rop erty that C e ′ ,p ∗ i ( p ) ≥ C e,p i ( p ) − D ∗ ≥ C − ( t + 1) D ∗ . Thus, th ere is at least one path p ′ ∈ p whic h is in the same buck et with p ∗ i (namely , B ( p ′ ) = B ( p ∗ i )) for whic h it holds that C e ′ ,p ′ ( p ) = C e ′ ,p ∗ i ( p ). Therefore, C e ′ ( p ) ≥ C − ( t + 1) D ∗ . Consequently , f rom th e deﬁn ition of f ( E t ) it follo ws that for any edge e ′ ∈ f ( E t ) it holds that C e ′ ( p ) ≥ C − ( t + 1) D ∗ . By consid er in g the un ion of E t ∪ f ( E t ), we ha v e that the claim h olds for any edge in e ∈ E t +1 , as needed. Lemma 4.5 In N ash-r outing p it holds tha t C ≤ 18 C ∗ · D ∗ · lg 2 n . Pro of: F rom Lemma 4.4, there is a set of edges X ⊂ Z w ith | f ( X ) | ≤ 2 | X | . F or eac h e ∈ Z it holds that C e ( p ) ≥ C − 2 D ∗ · lg n . Let Π = S e ∈ X Π e ( p ), that is, Π is the set of pla yers whic h in routing p their paths us e edges in X . Let M = P e ∈ X C e ( p ), which denotes the total “normalized” utilization of th e edges in X . W e h a v e that M ≥ | X | ( C − 2 D ∗ · lg n ). By construction, the congestion in routing p in eac h of the edges of X is caused only b y the pla ye rs in Π. W e can b oun d the path lengths of the pla y ers in Π with resp ect to D ∗ as follo ws. Consider a pla ye r i ∈ Π. W e ha ve th at C i ( p ) ≥ C − 2 ¯ D ∗ · lg n . F rom Lemma 4.3 and the fact that | s i | ≤ D ∗ , we obtain: | p i | ≤ | s i | + 2 D ∗ · lg n + 1 ≤ D ∗ + 2 D ∗ · lg n + 1 ≤ 4 D ∗ · lg n. T h u s , the path length of eve ry pla yer in Π is at most K = 4 D ∗ · lg n . M can also b e b ounded as M ≤ K | Π | . Consequently , | X | ( C − 2 D ∗ · lg n ) ≤ K | Π | , whic h giv es: C ≤ K | Π | | X | + 2 D ∗ · lg n = 4 D ∗ · lg n · | Π | | X | + 2 D ∗ · lg n. (1) Since | f ( e, i ) | ≥ 1, in the optimal routing p ∗ the path of eac h user in Π has to use at least one edge in f ( X ). Thus, in the optimal routing p ∗ , the edges in f ( X ) are used at least | Π | times. Thus, there 9 is some edge e ∈ f ( X ) which in p ∗ is u sed b y at least | Π | / | f ( X ) | paths. Sin ce th ere are lg L + 1 buc kets, the normalized congestion of e in one of those buc kets is at least | Π | / ( | f ( X ) | · (lg L + 1)). Therefore, C ∗ ≥ | Π | / ( | f ( X ) | · (lg L + 1)). Since | f ( X ) | ≤ 2 | X | , w e obtain: | Π | ≤ 2 C ∗ · | X | · (lg L + 1) ≤ 4 C ∗ · | X | · lg n. (2) By Combining Equations 1 and 2, w e get: C ≤ 16 C ∗ · D ∗ · lg 2 n + 2 D ∗ lg n ≤ 18 C ∗ · D ∗ · lg 2 n. When C > D / 4, using Lemma 4.5 it is straigh tforwa r d to prov e that P oA = O (( C ∗ · D ∗ · lg 2 n ) / ( C ∗ + D ∗ )). If C ≤ D / 4, then u sing Lemma 4.3, we can pr o v e th at P oA = O (1) (the details can b e found in the app endix). Therefore, we obtain the main r esult: Theorem 4.6 (Price of anarc hy in sum-buc k et games) F or any sum-bucket game R it holds: P oA = O C ∗ · D ∗ C ∗ + D ∗ · lg 2 n ! . There is a sum-b uc ke t game R = ( N , G, P ) that sho ws that th e result of Theorem 4.6 is tight in non-trivial cases. All the pla yers hav e th e same source n o de u and destination no d e v . L et a = √ N (supp ose f or simplicit y th at a = √ N = ⌈ √ N ⌉ ). There is a path p of length a fr om u to v . There are a edge-disjoin t paths Q = { q 1 , . . . , q a } f r om u to v so that eac h path q i has length a and uses one edge of p . Eac h pla y er has t wo paths in h er strategy set: one is path p and the other is a p ath fr om Q . F urther, for eac h path q i there are a play ers that ha v e q i in their strategy s ets. Let p b e the routing where ev ery pla ye r chooses path p . Then, p is a Nash-routing with so cial cost a + N . Let p ′ b e th e r outing w here eve ry pla yer chooses the alternativ e path in Q . Then, p ′ is also a Nash-routing with so cial cost 2 a . Thus, P oA ≥ S C ( p ′ ) /S C ( p ) ≥ ( a + N ) / ( a + 1) = Ω ( √ N ) = Ω( √ n ). R ou tin g p ′ is an optim um routing with the smallest so cial cost C ∗ = C ( p ) = a and D ∗ = D ( p ) = a . Thus, from Theorem 4.6 , P oA = O (( C ∗ · D ∗ · lg 2 n ) / ( C ∗ + D ∗ )) = O ( a · lg 2 n ) = O ( √ n · log 2 n ). Hence, the pr ice of anarc hy has to b e within a log 2 n factor from the b oun d pro vided in Theorem 4.6. 5 Conclusions In this work w e pro vided the ﬁrst study (to our knowledge ) of b icriteria routin g games, where the pla ye rs attempt to sim ultaneously optimize t wo parameters: their p ath congestion and length. The motiv ation is the existence of eﬃcien t pac ke t scheduling algorithms wh ic h d eliv er the pac ket s in time prop ortional to the so cial cost. W e examined max games and sum games. Max games stabilize, but their price of anarch y is h igh. S um games do not stabilize, b u t they can giv e b etter price of anarc hy . W e then giv e th e approximate sum-b uc ke t game whic h alw a ys s tabilizes and pr eserv e the go o d prop erties of s um games. S urprisin gly , arbitrary sum -buc ket game equilibria provi d e go o d appro ximations to the original co ord inated routing problem. Sev eral op en problems remain to examine. W e studied tw o particular functions of the bicriteria, namely , the max and the s um fu n ctions. There are other fu nctions, for example a weigh ted s u m, th at could pro vide s im ilar or b ette r r esu lts. It would also b e in teresting to add additional parameters. The original C + D sum games do not stabilize in general. Ho wev er, there exist interesti n g instances whic h stabilize . F or example, it can b e easily sho wn that the games where the a v ailable paths ha ve equal lengths alw a ys stabilize. It would b e interesting to ﬁn d a general c haracterization of the game instances that stabilize. Another in teresting problem is to pro vide time eﬃcie nt algorithms for ﬁnd ing equilibria in our games. 10 A Additional Pro ofs of Section 3.1 Lemma A.1 Minimum r outing p min achieves optimal so cial c ost, that i s, S C ( p min ) ≤ S C ( p ) , for any other r outing p 6 = p min . Pro of: Supp ose for con tradiction that there exists a routing p ≥ p min with S C ( p ) < S C ( p min ). Let S C ( p ) = max( C ( p ) , D ( p )) = k 1 , and S C ( p min ) = max( C ( p min ) , D ( p min )) = k 2 . Clearly , k 1 < k 2 . Th erefore, in the v ector M ( p ) = [ m 1 , . . . , m r ] it h olds that m k 1 6 = 0 and m k = 0 for k > k 1 . S imilarly , in the v ector M ( p min ) = [ b m 1 , . . . , b m r ] it h olds that b m k 2 6 = 0 and b m k = 0 for k > k 2 . T herefore, M ( p min ) > M ( p ), con tradicting the fact that p ≥ p min . B Additional Pro ofs of Section 4.1 In th e table b elo w we calculate the congestions, path lengths, and pla y er costs for eac h routing scenario of the game in Theorem 4.1. W e set the sp eciﬁc lengths as: | p 1 | = 10, | p ′ 1 | = 8, | p 2 | = 7, | p ′ 2 | = 10, | p 3 | = 7, and | p ′ 3 | = 10. F or a p la y er i ∈ { 1 , 2 , 3 } and routing p w e deﬁne the c omplementary r outing to b e the one where p la y er i c ho oses the alternativ e path. F or example, for pla yer 2 the complemen tary routing of [ p 1 , p 2 , p 3 ] is [ p 1 , p ′ 2 , p 3 ]. A play er is lo cally optimal in a routing if the complementary r outing d o es not give a low er cost for the pla yer. Using the table it is easy to determine wh ether a pla y er is lo cally optimal or n ot b y examining the resp ectiv e costs in the complementa r y routings. In this wa y , w e ﬁ nd th e non-lo cally pla y ers which are s h o wn in the righ tmost column of the table. routing C 1 D 1 pc 1 C 2 D 2 pc 2 C 3 D 3 pc 3 Non lo cally optimal pla yers [ p 1 , p 2 , p 3 ] 4 10 14 4 7 11 4 7 11 pla ye r 1 [ p ′ 1 , p 2 , p 3 ] 5 8 13 5 7 12 5 7 12 pla yer 2 [ p ′ 1 , p ′ 2 , p 3 ] 5 8 13 1 10 11 5 7 12 pla ye r 3 [ p ′ 1 , p ′ 2 , p ′ 3 ] 5 8 13 1 10 11 1 1 0 1 1 p la y er 1 [ p 1 , p ′ 2 , p ′ 3 ] 2 10 12 2 1 0 1 2 2 10 12 play er 2 [ p 1 , p 2 , p ′ 3 ] 3 10 13 4 7 11 2 1 0 1 2 p la y er 3 [ p 1 , p ′ 2 , p 3 ] 3 10 13 2 10 12 4 7 11 pla y er 2 [ p ′ 1 , p 2 , p ′ 3 ] 5 8 13 5 7 12 1 10 11 pla y er 2 C Additional Pro ofs of Section 4.2.1 Lemma C.1 If a g r e e dy move b y player i takes a r outing p to a new r outing p ′ , then p ′ < p . Pro of: Let p i and p ′ i denote the paths of i in routings p and p ′ , resp ectiv ely . Let pc i ( p ) = C i ( p ) + D i ( p ) = z 1 and pc i ( p ′ ) = C i ( p ′ ) + D i ( p ′ ) = z 2 . S ince play er i decreases its cost in p ′ , z 2 < z 1 . Consider now the vect ors of the r outings M ( p ) = [ m 1 , . . . , m r ] and M ( p ′ ) = [ m ′ 1 , . . . , m ′ r ]. W e will sho w that M ( p ) < M ( p ′ ). Let B ( p i ) = k 1 and B ( p ′ i ) = k 2 . Let V d enote the set of p la y ers w hose cost increases in p ′ with resp ect to their cost in p . Next, we show that for any j ∈ V it holds th at p c j ( p ′ ) ≤ c i ( p ′ ), whic h will help u s to pro ve the desired r esult. Let p j b e the path of j ∈ V . Let B ( p j ) = k 3 (note th at j do es not switc h paths an d buc ket s b et ween p and p ′ ). If k 3 6 = k 1 and k 3 6 = k 2 then the cost of j remains u naﬀected b et w een p and i p ′ . Th u s, either k 3 = k 1 or k 3 = k 2 . If k 3 = k 1 , then the cost of j can only decrease fr om p to p ′ , since path p i can no longe r aﬀect p ath p j . T herefore, it has to b e that k 3 = k 2 . Supp ose, for the sak e of contradict ion, that pc j ( p ′ ) > pc i ( p ′ ). Then, C j ( p ′ ) + D j ( p ′ ) > C i ( p ′ ) + D i ( p ′ ). Since b oth paths are in the s ame buc ket , D j ( p ′ ) = D i ( p ′ ), which implies that C j ( p ′ ) > C i ( p ′ ). Since j ∈ V , C j ( p ) < C j ( p ′ ). The increase in normalized congestion of p j in p ′ can only b e caused by p ′ i b ecause it sh ares an edge with p j with congestio n C j ( p ′ ). Ho wev er, this is imp ossible since it w ould im p ly that C i ( p ′ ) ≥ C j ( p ′ ). Th erefore, pc j ( p ′ ) ≤ pc i ( p ′ ). Consequent ly , in vect or p ′ all the en tries in p ositio ns j 2 + 1 , . . . , r d o not increase with r esp ect to p . F urth er, b ecause i s w itc hes p aths, m j 1 > m ′ j 2 . T h us, M ( p ) < M ( p ′ ), as n eeded. D Additional Pro ofs of Section 4.2.2 Lemma D.1 In N ash-r outing p , for every player i and e dge e it holds that | f ( e, i ) | ≥ 1 . Pro of: Supp ose that | f ( e, i ) | = 0. Then for ev ery edge e ′ ∈ p ∗ i it holds that C e ′ ,p ∗ i ( p ) < C e,p i ( p ) − D ∗ . Let C ′ = max e ′ ∈ p ∗ i C e ′ ,p ∗ i ( p ). If pla yer i was to choose path p ∗ i its cost would b e: pc ′ i ≤ C ′ + 1 + D p ∗ i < C e,p i ( p ) − D ∗ + 1 + D ∗ = C e,p i ( p ) + 1 ≤ C i ( p ) + D i ( p ) = pc i ( p ) . Thus, pc ′ i < pc i ( p ) whic h implies that p la y er i is not lo cally optimal in Nash-r ou tin g p , a con tradiction. Pro of of Theorem 4.6: Supp ose that p is the w orst Nash-routing with maxim u m so cia l cost. W e h a v e that P oA ≤ S C ( p ) /S C ( p ∗ ) ≤ ( C + D ) / ( C ∗ + D ∗ ) . W e examine t wo cases: • C > D / 4: In th is case D = O ( C ). F rom Lemma 4.5, C ≤ 18 C ∗ · D ∗ · lg 2 n . Therefore: P oA = O ( C / ( C ∗ + D ∗ )) = O (( C ∗ · D ∗ · lg 2 n ) / ( C ∗ + D ∗ )) . • C ≤ D / 4: Let p i ∈ p b e the path with maximum cost in p . Clearly , pc i ( p ) = C i + D i ≥ D . F urther, 0 ≤ C i ( p ) ≤ C ≤ D / 4. Th us, D i ≥ D − C i ≥ D − D / 4 = 3 D / 4. Sin ce C i ≥ 0 = C − C , Lemma 4.3 giv es | p i | ≤ | s i | + C + 1 ≤ D ∗ + D / 4 + 1. W e ha ve that D i / 2 < | p i | . T herefore, D i / 2 < D ∗ + D / 4 + 1, whic h giv es: 3 D / 8 < D ∗ + D / 4 + 1. Th u s, D < 8( D ∗ + 1) ≤ 16 D ∗ . In order wo r ds, D = O ( D ∗ ). Since, C = D , w e obtain: P oA = O ( D / ( C ∗ + D ∗ )) = O ( D ∗ / ( C ∗ + D ∗ )) = O (1) . By com bin ing the tw o ab ov e cases we obtain the desirable r esu lt. ii

Bicretieria Optimization in Routing Games

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